Houston Journal of Mathematics
c
University of Houston
Volume , No. ,
DISCRIMINATING AND SQUARELIKE GROUPS II:
EXAMPLES
BENJAMIN FINE
ANTHONY M. GAGLIONE
DENNIS SPELLMAN
This paper is dedicated to the memory of B.H. Neumann.
Abstract. Discriminating groups were invented by G. Baumslag, A.G.
Myasnikov and V.N. Remeslennikov in [BMR2]. These groups arose out
of consideration by the above authors of their freshly minted theory of algebraic geometry over groups. Although unforeseen at first, the notion admits
beautiful and exotic examples some of which appeared earlier in diverse contexts. Squarelike groups were invented by the authors in [FGMS2] and may
be viewed as nonstandard discriminating groups. In this paper we give examples of groups which are discriminating or squarelike as well as groups which
are not. The main result is the existence of a finitely generated squarelike
group which is not discriminating.
Contents
0. Introduction
1. Preliminaries
2. Groups Satisfying Various Finiteness Conditions and Abelian
Groups
3. Trivially Discriminating Groups
4. Nontrivially Discriminating Groups
5. Nondiscriminating Groups
6. Squarelike Groups
7. References
2000 Mathematics Subject Classification. Primary 20E26, Secondary 03C60.
Key words and phrases. discriminating group, squarelike group.
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3
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12
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BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
0. Introduction
A group G is called discriminating provided that given any finite set of nontrivial
elements S in G × G there exists a homomorphism f = fS : G × G → G such that
f (s) 6= 1 for all s ∈ S. (Note this is not the actual definition of discriminating
group but it is equivalent to it. See Definition 1.1 and Proposition 1.2.) In other
words, G is discriminating if G × G is fully residually G. In the special case where
G contains a subgroup isomorphic to G × G, then clearly G is discriminating and
we term such G trivially discriminating in Definition 1.3. A group G is called
squarelike provded that G and G × G have the same universal theory. It is not
hard to see that a discriminating group must be squarelike (see Proposition 1.4).
Even though the roots of this topic go back to classical work of B.H. Neumann
[N], it has only very recently become an increasingly active area of research. One
of the reasons for this is the relevance of these notions to the algebraic geometry over groups ( [BMR1] and [BMR2]) and the recent breakthrough progress
by Sela ([Se1]) and Kharlampovich-Myasnikov ([KhM]) on the celebrated Tarski
Conjecture and on understanding the elementary theory of free groups. The
Tarski Conjecture basically says that all the non-abelian free groups satisfy the
same first-order sentences. Furthermore an expository treatment of the elementary theory of free groups can be found in the authors’ paper [FGMS3]. Another
reason for this renewed interest is that, as it turns out, there are some very
interesting classes of groups (particularly, the so-called ”branch” and automata
groups acting on rooted trees, including the famous Grigorchuk group as well as
the finitely generated infinite simple groups of G. Higman, including one of the
famous Thompson groups) that provide natural sources of examples of discriminating groups.
Since the subject remains fairly new, the present paper serves to introduce it
to a wider audience. While the paper is mainly expository, it also does contain
some new results.
The paper is divided into six sections. In Section 1 we fix definitions and
notation as well as develop the material from mathematical logic needed later
in the paper. In Section 2 we consider various finiteness conditions and their
effects on discrimination and being squarelike or the relationship between the
two. We also give a partial characterization of abelian discriminating groups due
to Baumslag, Myasnikov and Remeslennikov which first appeared in [BMR2] as
well as a complete characterization of abelian squarelike groups due to the authors
which first appeared in [FGMS2]. In Section 3 we give examples of groups which
embed their direct squares. In Section 4 we give examples of discriminating groups
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
3
which do not embed their direct squares. In Section 5 we give examples of groups
which are not discriminating. In Section 6 we give axioms due to Verena HuberDyson for the class of squarelike groups. The authors had previously proven
that this class is axiomatic, but an explicit axiom schema is new. We refer to
[FGS2] where we actually prove that the class of squarelike groups is the model
class of these axioms together with the group axioms. Section 6 also contains an
example of a finitely generated group which is squarelike but not discriminating.
Previously the only known squarelike, nondiscriminating groups were not finitely
generated.
1. Preliminaries
We begin by fixing notation and terminology. Here N shall be the set of positive
integers. If I is a nonempty index set and (Gi )i∈I is a family of groups indexed by
I, then the unrestricted direct product of the family, Πi∈I Gi , shall be the set of
all choice functions I → ∪i∈I Gi with multiplication defined componentwise. The
bnotconc(b)TjΩ5.7ordTjΩ88.9592
0 TΩ(wactually5178
0T
restricted direct product of the family shall be the subgroupwiof
squarelithos3.716
h are
-2-1.471 TdΩ(whic)TjΩ9338
0 TdΩ(hoice)TjΩ25
4
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
Thus a sufficient (but, as we shall see, not necessary) condition that G be
discriminating is that G × G embed in G.
Definition 1.3. The group G is trivially discriminating provided G×G embeds in G.
In order to introduce squarelike groups we first discuss some preliminaries from
logic. Let L be the first-order language with equality containing a binary operation symbol •, a unary operation symbol −1 and a constant symbol 1. Being
first-order means that in every interpretation the variables are viewed as roaming
over individual group elements only but never over subsets nor functions. Moreover, = is the only relation symbol and is always to be interpreted as the identity
relation. A quantifier-free formula of L is equivalent to one in disjunctive normal form


^
_ ^
 (Aij = aij ) ∧ (Bik 6= bik )
j
i
k
as well as one in conjunctive normal form


_
^ _
 (Cij = cij ) ∨
(Dik 6= dik ) .
j
i
k
(We use 6= as an abbreviation for ˜( = ).) Here the Aij , aij , Bik , bik , Cij , cij , Dik
and dik are terms of L. Assuming the group axioms
∀x, y, z ((x • y) • z = x • (y • z)) ,
∀x (x • 1 = x) ,
∀x x • x−1 = 1
these may be simplified to




_ ^
^
^ _
_
 (αij = 1) ∧ (βik 6= 1) and  (γij = 1) ∨ (δik 6= 1)
i
j
k
i
j
k
respectively, where αij , βik , γij and δik are words on the variables and their formal
inverses.
A universal sentence of L is one of the form ∀xϕ(x) where x is a tuple of
distinct variables and ϕ(x) is a quantifier-free formula of L containing at most
the variables in x. Similarly, an existential sentence of L is one of the form
∃xϕ(x) where x and ϕ(x) are as before. Vacuous quantifications are permitted
and a quantifier-free sentence of L is considered a special case both of universal
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
5
sentences and existential sentences. Observe that the negation of a universal
sentence is logically equivalent to an existential sentence and vice-versa. If G is a
group, then the universal theory of G, written Th∀ (G), is the set of all universal
sentences of L true in G. Two groups G and H are universally equivalent
or have the same universal theory, written G ≡∀ H, provided Th∀ (G) =
Th∀ (H). Observe that two groups satisfy precisely the same universal sentences
if and only if they satisfy precisely the same existential sentences. A practical
criterion for deciding the universal equivalence of two groups involves a special
kind of existential sentence. The existential sentence


^
^
∃x  (Aj = aj ) ∧ (Bk 6= bk )
j
k
where the Aj , aj , Bk and bk are terms of L is primitive. In the event the group
axioms are assumed the above may be simplified to


^
^
∃x  (αj = 1) ∧ (βk 6= 1)
j
k
where the αj and βk are words on the variables and their formal inverses.
Now suppose that G and H are groups and G ≤ H. Then, since universal
properties are inherited by substructures, every universal sentence of L true in H
must also be true in G. But, in general, G will satisfy more universal sentences
than H. (Equivalently, H will satisfy more existential sentences than G.) So,
in the event G ≤ H, in order to show that G ≡∀ H it suffices to show that
every existential sentence true in H must also be true in G. Let ∃xϕ (x) be an
existential sentence of L with its matrix ϕ (x) written in disjunctive normal form.
Assuming the group axioms, ϕ (x) is ∨i ϕi (x) where ϕi (x) is
^
^
(wij (x) = 1) ∧ (uik (x) 6= 1).
j
k
But ∃x(∨i ϕi (x)) is logically equivalent to ∨i ∃xϕi (x) and a disjunction will be
true if and only if at least one of the disjuncts is true. Thus, in the event G ≤ H,
a necessary and sufficient condition for G ≡∀ H is that every primitive sentence
of L true in H be also true in G. That, in turn, reduces to the criterion that
every finite system
wi (x1 , ..., xk ) = 1, 1 ≤ i ≤ I
uj (x1 , ..., xk ) 6= 1, 1 ≤ j ≤ J
6
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
of equations and inequations which has a solution in H also have a solution in G.
We immediately deduce –
Proposition 1.4. ([FGMS1]): If G is discriminating, then G×G ≡∀ G.
Definition 1.5. ([FGMS2]): The group G is squarelike provided G×G ≡ ∀ G.
Hence, every discriminating group is squarelike (but, as we shall see, not conversely).
The dual of a proper ideal in a Boolean algebra is a filter. Specifically, if I is
a nonempty set and D is a family of subsets of I, then D is a filter on I provided
the following four conditions are satisfied:
(1)
(2)
(3)
(4)
φ∈
/ D;
I ∈ D;
A ∩ B ∈ D whenever A, B ∈ D;
B ∈ D whenever A ∈ D and A ⊆ B ⊆ I.
The trivial filter {I} is a filter on I. If I is infinite, the family
{A ⊆ I : I \ A is finite}
of cofinite subsets of I is a filter on I. Let I be a nonempty set and (Gi )i∈I
be a family of groups indexed by I. Let P = Πi∈I Gi be the unrestricted direct
product of the family (Gi )i∈I . Suppose D is a filter on I. Then the relation on
P defined by
f ≡D g iff {i ∈ I : f (i) = g(i)} ∈ D
is a congruence on P . Hence, KD = {f ∈ P : f ≡D 1} is a subgroup normal in
P . In this context we give:
Definition 1.6. The quotient group P/KD is the reduced product or filter
product of the family (Gi )i∈I modulo the filter D on I.
Observe that if D = {I} is the trivial filter, then KD = 1; hence, (up
to isomorphism) unrestricted direct products are instances of reduced products.
Observe also that, if I is infinite and D is the filter of cofinite subsets of I,
then KD is the restricted direct product of the family (Gi )i∈I . In the case of
unrestricted direct powers the diagonal map δ : G → GI , δ(g)(i) = g for all
i ∈ I, embeds G in GI . In that event, if D is a filter on I, we get a well-defined
monomorphism d : G → GI /KD determined by d(g) = δ(g)KD . Thus d is the
canonical embedding of G into the reduced power GI /KD .
We now discuss those sentences which are preserved in reduced products. A
quantifier-free formula ϕ of L is a Horn formula of L provided it is in conjunctive
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
7
normal form ∧i ∨j ϕij where disjunct ϕij is either atomic Aij = aij or negated
atomic Aij 6= aij ( Aij and aij are terms) and in each conjunct (i.e. for each
fixed i) at most one disjunct ϕij is atomic. Assuming the group theory axioms,
we may take an atomic formula to be one of the form α = 1 where α is a word on
the variables and their formal inverses. A Horn sentence of L is obtained by
appending quantifiers to the left of an unquantified Horn formula until no variable
remains free in the formula. Horn sentences are preserved in reduced products.
The converse is also true. That is, a sentence is preserved in arbitrary reduced
products if and only if it is logically equivalent to a Horn sentence. See e.g. [G].
Observe that the universal sentence ∀x (∧i (Pi (x) = pi (x)) → (Q(x) = q(x))) is
equivalent to the Horn sentence ∀x ((∨i (Pi (x) 6= pi (x)) ∨ (Q(x) = q(x))) so is
preserved in reduced products.
The sentence ∀x (∧i (Pi (x) = pi (x)) → (Q(x) = q(x))) is called a quasi-identity.
In the presence of the group axioms we may simplify the above to ∀x ∧i (wi (x) =
1) → (u(x) = 1)) where u(x) and the wi (x) are words on the variables and their
formal inverses. Observe that every identity ∀x(A(x) = a(x)) is equivalent to
a quasi-identity ∀x((1 = 1) → (A(x) = a(x))). So laws and the group axioms
themselves may be viewed as instances of quasi-identities.
Definition 1.7. The model class of a set of quasi-identities including the group
axioms is a quasivariety of groups.
For example, the class of torsion-free groups is a quasivariety being the model
class of the group axioms and the sentences (one for each integer n ≥ 2) ∀x((xn =
1) → (x = 1)). Quasivarieties are examples of axiomatic classes. That is,
they are model classes of sets of sentences of L. Quasivarieties are closed under
(unrestricted) direct products since they have a set of Horn axioms. They are also
closed under subgroups since they have a set of universal axioms. Put another
way, quasivarieties are axiomatic prevarieties, and, in fact, may be characterized
as such. See e.g. [C]. The class of all groups is a quasivariety of groups and
the intersection of any family of quasivarieties is again a quasivariety. For that
matter, the intersection of any family of universally axiomatizable model classes
(i.e. axiomatic classes having a set of universal axioms) is again a universally
axiomatizable model class. It follows that if X is any class of groups there is
a least quasivariety qvar(X ) containing X and a least universally axiomatizable
model class ucl(X ) containing X . Thus qvar(X ) is the quasivariety generated by
X and ucl(X ) is the universal closure of X . If X = {G} is a singleton we write
qvar(G) and ucl (G) for qvar(X ) and ucl(X ) respectively. The qvar(G) is the
model class of the quasi-identities satisfied by G and the ucl(G) is the model class
8
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
of the universal sentences satisfied by G. In general ucl(G) is a proper subclass
of qvar(G). (The more properties a structure is required to satisfy the fewer
structures can satisfy them in general.)
If G is a group then the first-order theory of G, written Th(G), shall be the
set of all sentences of L true in G. Two groups G and H are elementarily
equivalent, written G ≡ H, provided Th(G) = Th(H). A necessary condition
that a class of groups be axiomatic is that it be closed under elementary equivalence. (One can show, for example, that the class of finite groups is closed under
elementary equivalence but is not axiomatic; so, the condition is not sufficient.
Necessary and sufficient conditions are known, but we shall not give them here.
See e.g. [G].)
We conclude the preliminaries with a brief discussion of algebraic geometry
over groups. The theory is most useful in its relativized version where a group Γ
plays the role of the ring of scalars. For our purposes we need consider the case Γ
= 1 only so that a Γ-group G is just a group. Let n ∈ N and let Xn = {x1 , . . . , xn }
be a set of n distinct variables. Let S be a set of words on Xn ∪ Xn−1 . If G is
a group, then by a solution in G to the system S = 1 is meant an ordered ntuple g = (g1 , . . . , gn ) ∈ Gn such that s(g1 , . . . , gn ) = 1 for all s(x1 , . . . , xn ) ∈
S. We write S(g) = 1 in that event. The solution set of the system is then
VG (S) = {g ∈ Gn : S(g) = 1}. These are the affine algebraic subsets of
Gn and they form a closed subbase for a topology (the Zariski topology) on
Gn . The Zariski topology will be Noetherian just in case every descending chain
A0 ⊇ A1 ⊇ . . . ⊇ An ⊇ . . . of closed sets stabilizes after finitely many steps
AN = AN +1 = . . . . A necessary and sufficient condition that the Zariski topology
on Gn be Noetherian for all n is given in the following –
Definition 1.8. The group G is equationally Noetherian provided for every
n ∈ N and every system S = 1 of equations in n unknowns there is a finite subset
S0 ⊆ S such that VG (S) = VG (S0 ).
Examples of equationally Noetherian groups are groups linear over a commutative Noetherian ring with 1. In particular, every group linear over a field is
equationally Noetherian. Also according to Theorem 1.22 of [Se2], torsion-free
word-hyperbolic groups are equationally Noetherian.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
9
2. Groups Satisfying Various Finiteness Conditions and Abelian
Groups
Lemma 2.1. Suppose the set of nontrivial elements of finite order in the group G
is finite and nonempty. Then G is not squarelike and hence not discriminating.
Proof: Suppose G contains exactly n elements of finite order where n > 1 is
an integer. Let the least common multiple of the orders of these elements be m.
Then G satisfies
^
_
∀x1 , . . . , xn+1 (
(xm
(xi = xj ))
i = 1) →
1≤i≤n+1
1≤i<j≤n+1
but G × G does not.
We immediately deduce that no nontrivial finitely generated nilpotent group
with torsion can be squarelike. In particular, no nontrivial finitely generated
abelian group with torsion and no nontrivial finite group can be squarelike. What
about torsion free abelian groups?
Lemma 2.2. ([FGMS1]) Every torsion free abelian group is discriminating. Hence,
every torsion free abelian group is squarelike.
Proof: Let A be a torsion free abelian group. We write A additively. The
proof will proceed by induction on n. For n = 1, since A separates A ⊕ A, given
a1 ∈ (A ⊕ A)\{0}, there is f1 ∈ Hom(A ⊕ A, A) such that f1 (a1 ) 6= 0. Suppose
that, given a1 , . . . , an ∈ (A ⊕ A)\{0}, there is fn ∈ Hom(A ⊕ A, A) such that
fn (ai ) 6= 0, i = 1, . . . , n. Now suppose a1 , . . . , an+1 ∈ (A ⊕ A)\{0}. Then, by
inductive hypothesis, there are f, g ∈ Hom(A ⊕ A, A) such that f (ai ) 6= 0, i =
1, . . . , n, and g(an+1 ) 6= 0. We then choose an integer N sufficiently large so that
h = f +N g does not annihilate any of a1 , . . . , an+1 . That completes the induction
and proves the lemma. So, among finitely generated abelian groups, the discriminating groups and
the squarelike groups are precisely the torsion free ones. A moment’s reflection
produces nontrivial discriminating abelian groups ( necessarily not finitely generated) with torsion. For example, every group free of infinite rank in the variety
of abelian groups of exponent dividing n for any fixed integer n > 1 is trivially
discriminating. At this point three questions naturally arise.
(1) Can we characterize abelian discriminating groups?
10
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
(2) Can we characterize abelian squarelike groups?
(3) What, if anything, can we say about finitely generated torsion free nilpotent
groups?
A partial answer to (1) was given in [BMR2]. In order to discuss this and
other matters to follow, we must introduce the Szmielew invariants [S] of an
abelian group. Given an integer m > 0 and a family of elements (ai ) in an
additively written abelian group A, (ai ) is linearly independent modulo m
provided Σi ni ai = 0 implies ni ≡ 0(mod m) for all i; (ai ) is strongly linearly
independent modulo m provided Σi ni ai ∈ mA implies the coefficients ni ≡
0(mod m) for all i. For each prime p and positive integer k we define three ranks
each of which is a nonnegative integer or the symbol ∞.
• The maximum number of elements of A of order pk and linearly independent modulo pk is ρ(1) [p, k](A).
• The maximum number of elements of A which are strongly linearly independent modulo pk is ρ(2) [p, k](A).
• The maximum number of elements of A of order pk and strongly linearly
independent modulo pk is ρ(3) [p, k](A).
Proposition 2.3. (Szmielew [S]) Let A and B be abelian groups. Then A and B
are elementarily equivalent if and only if the following two properties are satisfied.
(1) The groups A and B either both have finite exponent or both have infinite
exponent.
(2) For all primes p and positive integers k, ρ(i) [p, k](A) = ρ(i) [p, k](B)) for
i = 1, 2, 3.
For a discussion of abelian groups we refer the reader to Kurosh [K, Vol. I].
Proposition 2.4. ([BMR2]) Let A be a torsion abelian group such that, for each
prime p, the p-primary component of A modulo its maximal divisible subgroup
contains no nontrivial elements of infinite p-height. Then A is discriminating if
and only if, for each prime p, the following two properties are satisfied.
(1) For all positive integers k, ρ(1) [p, k](A) is either 0 or ∞.
(2) The rank of the maximal divisible subgroup of the p-primary component of
A is either 0 or infinite.
We have a complete characterization of abelian squarelike groups.
Proposition 2.5. ([FGMS2]) Let A be an abelian group. Then A is squarelike
if and only if, for each prime p and positive integer k, ρ(1) [p, k](A) is either 0 or
∞.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
11
These results allow us to construct non-finitely generated groups which are
squarelike but not discriminating. The following explicit example was given in
[FGMS2].
Example 1. For each positive integer n, let Mn be a free Z/(2n ) –module of
countably infinite rank. Let M be the direct sum ⊕n Mn of the Mn (i.e. restricted
direct product) as abelian groups, Let D be a rank 1 divisible abelian 2-group.
Then A = M ⊕D is squarelike but not discriminating. Note that, from Szmielew’s
criteria, A ≡ M . Moreover, M is trivially discriminating. It follows that the class
of discriminating groups is not closed under elementary equivalence – hence not
axiomatic.
Returning to finitely generated nilpotent groups, let us observe that they
are finitely presentable.
Proposition 2.6. ([FGMS2]) A finitely presented squarelike group is discriminating.
Proof: Let ha1 , . . . , am ; R1 (a1 , . . . , am ) = . . . = Rn (a1 , . . . , am ) = 1i be a finite
presentation of the squarelike group G. Then
ha1 , . . . , am , b1 , . . . , bm ;
Ri (a1 , . . . , am ) = Ri (b1 , . . . , bm ) = 1, 1 ≤ i ≤ n, [ai , bj ] = 1, 1 ≤ i, j ≤ mi
is a presentation for G × G. Let a = (a1 , . . . , am ), b = (b1 , . . . , bm ) and let gj
be nontrivial elements of G × G, j = 1, . . . , k. Write gj as a word gj = wj (a,b).
Then G × G satisfies the existential sentence σ:
^
∃x,y(
((Ri (x) = 1) ∧ (Ri (y) = 1)) ∧
1≤i≤n
^
1≤i,j≤m
([xi , yj ] = 1) ∧
^
(wj (x, y) 6= 1)))
1≤j≤k
Since G × G and G satisfy precisely the same universal sentences, they must also
satisfy precisely the same existential sentences; so, σ is also true in G. Let c =
(c1 , . . . , cm ) and d = (d1 , . . . , dm ) be tuples in G which verify σ upon substitution
for x and y respectively. Then the map on the generators ai 7→ ci , bi 7→ di , i =
1, . . . , m, preserves the relations so extends to a homomorphism ϕ: G × G → G.
But then ϕ(gj ) = wj (c,d) 6= 1 for all j = 1, . . . , k since x = c, y = d verifies σ
in G. Alternatively, since finitely generated torsion free nilpotent groups are linear
over a field, we could apply –
12
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
Proposition 2.7. ([FGMS1]) A squarelike, finitely generated, equationally Noetherian group is discriminating.
So, for finitely generated nilpotent groups, the notions of squarelike and discriminating coincide.
Proposition 2.8. ([BFGS]) If a finitely generated nilpotent group is discriminating, then it is abelian.
The best results in this direction to date are –
Proposition 2.9. ([MS]) If the finitely generated group G is linear over a field
and discriminating, then it is abelian.
Proposition 2.10. ([MS], [Ka]) If the finitely generated solvable group G is
discriminating, then it is abelian.
So, the only finitely generated discriminating linear or solvable groups are
torsion free abelian. At this point another question naturally arises besides the
three mentioned previously.
(4 ) Must every finitely generated squarelike group be discriminating?
In the next section (on trivially discriminating groups) we shall present an
example first exhibited in [FGS2] of a finitely generated discriminating group
which is neither equationally Noetherian nor finitely presentable.
There is our previous experience with universally free groups to draw
upon. A group is universally free just in case it has the same universal theory as
a nonabelian free group. A finitely generated group G is universally free if and
only if it is nonabelian and fully residually free in the sense of being discriminated
by a free group. By way of analogy, one might expect every finitely generated
squarelike group to be discriminating. But that turns out to be false. We present
a counterexample in our final section.
3. Trivially Discriminating Groups
Suppose G is any group whatsoever and I is an infinite index set. Then
GI × G I ∼
= GI so GI is surely trivially discriminating. If G 6= 1, then GI is
uncountable. Nontrivial countable examples of groups isomorphic to their direct
squares are easy enough to construct. Just take the restricted direct power of a
nontrivial countable group with respect to a countably infinite index set. What
about examples of nontrivial finitely generated groups isomorphic to their direct
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
13
squares? These exist! Such examples were first constructed by Tyrer Jones [J]
and subsequently by Hirshon and Meier [HM]. The question of whether or not
there exists a nontrivial finitely presented group isomorphic to its direct square
remains open. None the less, if we relax the condition G × G ∼
= G to G × G
embeds in G, finitely presented such groups G 6= 1 do exist. As G. Baumslag
observed we have –
Proposition 3.1. ([BFGS]) A simple discriminating group is trivially discriminating.
Example 2. For each integer n ≥ 2 and each integer r ≥ 1, Higman in [Hi2] defined a group Gn,r . Let Vn be the variety of all algebras with one n-ary operation
λ and n unary operations α1 , . . . , αn subject to the laws λ(α1 (x), . . . , αn (x)) = x
and αi (λ(x1 , . . . , xn )) = xi , i = 1, . . . , n. If Vn,r is an algebra free on r generators in Vn , then we let Gn,r = Aut(Vn,r ) be its group of automorphisms. Higman
proved that the Gn,r are finitely presented, when n is even Gn,r is simple and when
+
n is odd Gn,r contains a simple subgroup G+
n,r of index 2. Setting Gn,r = Gn,r
∼ +
when n is even, he showed that, for fixed r, G+
m,r = Gn,r implies m = n. Thus
he found an infinite family of finitely presented infinite simple groups. Higman
remarks that one of the groups in the family G+
n,r was discovered by Richard
Thompson, whose name we shall repeat soon. The Gn,r are discriminating; so,
when n is even, Gn,r is a finitely presented trivially discriminating group by
Proposition 3.1.
We show directly, regardless of the parity of n, that Gn,r is trivially discriminating. To do so we repeat here Higman’s observations that if X is a set of
free generators for an algebra V in Vn then, for each x ∈ X, so is (X\{x}) ∪
{α1 (x), . . . , αn (x)} and hence r ≡ s(mod(n − 1)) implies Vn,r ∼
= Vn,s . Let us fix
n ≥ 2 and r ≥ 1. Let s ≥ 2r be such that s ≡ r(mod(n − 1)). Let {x1 , . . . , xs } be
an s element set of free generators for Vn,r . For every ordered pair of automorphisms (θ,ϕ) of Vn,r we define an automorphism ψ of Vn,r by
ψ(xi ) = θ(xi ), i = 1, . . . , r
ψ(xi ) = ϕ(xi ), i = r + 1, . . . , 2r
ψ(xi ) = xi , i = 2r + 1, . . . , s
where θ acts on the free algebra hx1 , . . . , xr i ∼
= Vn,r and ϕ acts on the free algebra
hxr+1 , . . . , x2r i ∼
= Vn,r . Clearly the assignment (θ, ϕ) 7→ ψ is an embedding
Gn,r × Gn,r ,→ Gn,r . Thus the Gn,r are all trivially discriminating.
14
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
Example 3. The group F , due to Richard Thompson, of all orientation preserving piecewise linear homeomorphisms from the unit interval [0,1] onto itself that
are differentiable except at finitely many dyadic rational numbers and such that
on intervals of differentiability the derivatives are powers of 2 is finitely presented.
See [CFP]. It is proven in [FGMS1] that F is trivially discriminating.
Example 4. For each odd prime p, Gupta and Sidki in [GS] constructed a group
Hp which is a subgroup of the group of automorphisms of a rooted tree. The group
Hp is a 2-generator infinite p-group. Consider the commutator subgroup Hp0 of
Hp . It can be shown that Hp0 , while finitely generated, is not finitely presentable.
Moreover, it can be shown that Hp0 is trivially discriminating.
Let us dub a finitely presented group G a universal finitely presented
group provided it embeds every finitely presented group.
Example 5. Higman proved the existence of such groups in [Hi1]. Clearly
every universal finitely presented group is trivially discriminating. It is shown in
[BFGS] that if U is a recursively presentable abelian group and T is a universal
finitely presented group, then the standard wreath product U wrT is trivially
discriminating. On the other hand, it is shown in that same paper that if U is a
nonabelian finitely presented group and T is a universal finitely presented group,
then U wrT is not discriminating let alone trivially so.
Example 6. The existence of a finitely generated but not finitely presentable
trivially discriminating group G1 (e.g. the commutator subgroup of a GuptaSidki group) together with the existence of a universal finitely presented group
G2 allowed us to construct in [FGS2] a group G = G1 × G2 which is proven in
that paper to be a finitely generated discriminating group which is neither finitely
presented nor equationally Noetherian.
Example 7. The group Sω of all permutations of N which move only finitely
many integers and the subgroup Aω of even permutations are each trivially discriminating as we shall presently show; moreover, Sω embeds in Aω . Given an
integer n ≥ 2, we define maps fi : Sn → S2n , i = 1, 2, and g : Sn → A2n as
follows
f1 (π)(2k − 1) = 2π(k) − 1, f1 (π)(2k) = 2k; k = 1, . . . , n.
f2 (π)(2k − 1) = 2k − 1, f2 (π)(2k) = 2π(k); k = 1, . . . , n.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
15
Then f : Sn × Sn → S2n defined by f (π1 , π2 ) = f1 (π1 )f2 (π2 ) is easily seen to
be an embedding. We define g by g(π) = f (π, π). Observe that sgn f (π1 , π2 )
= sgn(π1 )sgn(π2 ) so that f restricts to an embedding An × An ,→ A2n and,
moreover, the image of g is indeed contained in A2n . Since Sω and Aω are the
direct limits
Sω = limn≥2 Sn and Aω = limn≥2 An ,
−−→
−−→
we get embeddings Sω × Sω ,→ Sω , Aω × Aω ,→ Aω and Sω ,→ Aω . Thus we have
proven all the claims above.
Let F be the class of all finite groups. Then, since every finite group embeds
in Sω and in Aω , every universal sentence true in Sω or in Aω must be true in
every finite group. On the other hand, universal sentences are preserved in direct
unions and each of Sω and Aω is a direct union of finite groups. It follows that
Th∀ (F) = Th∀ (Sω ) = Th∀ (Aω ) where Th∀ (F) is the set of all universal sentences
of L true in every finite group.
It is worth mentioning that, if G is any trivially discriminating group,
an easy induction shows that Gn embeds in G for every positive integer n. Conversely, if Gn embeds in G for some integer n ≥ 2, then G is trivially discriminating
since clearly, in that event, G × G ≤ Gn .
4. Nontrivially Discriminating Groups
Our first example of this section is a consequence of Lemma 2.2.
Example 8. Let r > 0 be an integer. Let G be free abelian of rank r. No
subgroup of G can have rank greater than r; hence, G × G of rank 2r cannot
embed in G. Thus, G is a finitely generated nontrivially discriminating group.
It is easy to see that the class of discriminating groups is closed under direct
products. (It is even closed under reduced products. See [FGMS2].)
Lemma 4.1. Let G be any group.
(A) If G contains an element of infinite order but no subgroup free abelian of
rank 2, then G × G cannot embed in G.
(B) If G is torsion by cyclic, then G cannot contain any subgroup free abelian
of rank 2.
(C) If G is an extension of a torsion group by an infinite cyclic group, then
G × G cannot embed in G.
16
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
Proof: Part (A) is obvious and (C) follows from (A) and (B). Thus it suffices to
show (B). Assume G is torsion by cyclic. Observe that, if the commuting elements
b1 and b2 of G were linearly independent over Z, then for all (p, q) ∈ Z2 \{(0, 0)}
and all n ∈ N, (bp1 bq2 )n has infinite order so is nontrivial modulo any normal torsion
subgroup. Thus, bp1 bq2 would be nontrivial in the image for all (p, q) ∈ Z2 \{(0, 0)}.
But no cyclic group can contain a subgroup free abelian of rank 2. Example 9. Letting G = Hp0 × C where Hp0 is the commutator subgroup of a
Gupta-Sidki group and C is infinite cyclic we see (recalling that Hp0 is a p-group
– hence torsion) from Lemma 4.1 that the nonabelian discriminating group G
cannot be trivially discriminating.
Are there any nonabelian examples which cannot be constructed in the facile
manner of taking direct products with finitely generated free abelian groups?
Here are some possibilities.
Example 10. Let p be a prime and let Ω : N → {0, 1, . . . , p} be an infinite
sequence of integers 0, 1, . . . , p. For each such sequence Grigorchuk [Gr] defined a
finitely generated group GΩ which has intermediate growth. These groups have
the following two properties.
(1) GΩ is residually a finite p-group for every sequence Ω.
(2) If every number from the set {0, 1, . . . , p} occurs infinitely many times, then
GΩ contains a copy of every finite p-group as a subgroup.
It was proven in [FGMS1] that, if Ω contains every integer from {0, 1, . . . , p}
infinitely many times, then GΩ is discriminating. The proof given there did not
produce an embedding GΩ × GΩ ,→ GΩ . So conceivably, one or more of these GΩ
is nontrivially discriminating. As of the writing of this paper, we do not know.
None the less, Peter M. Neumann has come to our rescue. He pointed out
that the groups in a (uncountably infinite!) family of finitely generated groups
discovered by his father [N] are discriminating but not trivially so.
Example 11. Let n = (n(1), n(2), . . . , n(r), . . .) be a strictly increasing sequence
of odd positive integers with n(1) ≥ 5. Fix n and let G = Gn be the subgroup
of the unrestricted direct product Πr∈N An(r) of the alternating groups An(r) of
degree n(r) generated by two elements x and y where, for each r ∈ N, the rth coordinates of x and y are given by x(r) = (123) and y(r) = (12 . . . n(r))
respectively. Let g1 , . . . , gm be nontrivial elements of G × G. Since G is residually
finite there is a finite homomorphic image G0 of G×G in which none of g1 , . . . , gm
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
17
is annihilated. Then, for sufficiently large r, G0 embeds in An(r) and An(r) ≤
K0 ≤ G where K0 is the restricted direct product of the family (An(r) )r∈N . (For
a proof that K0 is a subgroup of G see [K, Vol. II] or [N].) Hence, there is a
homomorphism
ϕ:G×G→G
for which ϕ(gi ) 6= 1, i = 1, . . . , m. Thus G is discriminating.
We show that G is not trivially discriminating by proving it is torsion by cyclic
and applying Lemma 4.1. Let T be the normal closure of hxi in G, here denoted
by hxiG . For each integer k ≥ 0 a product
z = y m(1) xε(1) y −m(1) . . . y m(k) xε(k) y −m(k) of k conjugates of x±1 is such that,
for each r ∈ N, at most 3k integers are moved by the r-th coordinate z(r) of z.
Consequently, for all r ∈ N, we can view z(r) as lying in an isomorphic copy of
the symmetric group S3k . Thus, z (3k)! = 1 and this is clearly sufficient for T to be
a torsion subgroup of G. Since y has infinite order, no positive power of y can lie
in the torsion subgroup T . Clearly, G/T is infinite cyclic generated by the image
of y modulo T . We have therefore established that the finitely generated groups
G = Gn of B.H. Neumann are not trivially discriminating.
The groups GΩ of Grigorchuk are not finitely presentable and we do not know
whether or not any of the groups Gn of B.H. Neumann are finitely presentable.
Peter M. Neumann has also found finitely presented, nonabelian nontrivially discriminating groups.
Example 12. Let X be a nonabelian, finitely generated, torsion free nilpotent
group of class 2 and let Y be one of the infinite simple groups Gn,r (with n even)
of Graham Higman discussed in Example 2 of the previous section. ([B] is a
reference for all properties of X and [Hi2] is a reference for all properties of Y .)
The properties of X and Y needed are
(a) X and Y are finitely presented.
(b)X is residually finite.
(c) Every finite group is embeddable in Y .
(d) X is not embeddable in Y ; in fact a finitely generated nil-2 subgroup of Y
is finite by abelian.
(e) Y × Y × Y is embeddable in Y .
Peter’s finitely presented examples are the groups G = X × Y . Clearly G is
nonabelian and finitely presented. To show that G is discriminating it suffices to
show that Y discriminates any group of the form W ×Y ×Y where W is residually
18
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
finite. Let h1 , . . . , hn be finitely many nontrivial elements of W × Y × Y . Write
hi = (ai , bi ) where ai ∈ W and bi ∈ Y × Y . Without loss of generality we may
assume ai 6= 1 for i = 1, . . . , m and ai = 1 for i = m + 1, . . . , n. Since W is
residually finite there exists a finite group V and a homomorphism α : W → V
such that α(ai ) 6= 1 for i = 1, . . . , m. By (c) above we may embed V in Y and so we
get a homomorphism β : W → Y such that β(ai ) 6= 1 for i = 1, . . . , m. Let γ be
the homomorphism γ : W ×Y ×Y → Y ×Y ×Y defined by γ(w, y, z) = (β(w), y, z).
Let δ be an embedding of Y × Y × Y into Y , which exists by (e) above. If we let
ϕ = δ ◦ γ then ϕ maps W × Y × Y into Y and ϕ(hi ) 6= 1 for all i = 1, . . . , n. Thus
G584ψ0ψTd (in)Tj 8.03894ψ0/R48ψ9.96Ccriminate(us)Tj /R48ψ9.96264ψT6 10..755ψ0ψTd (Y)Tj /R54ψ9.96264ψTf 10.0708ψ0ψTd ()Tj /R
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
19
Proposition 5.2. The free product of two nondiscriminating groups is nondiscriminating.
Proof: Let A and B be nondiscriminating groups. We may assume that A is
not CT and, in particular, nonabelian. Suppose a1 and a2 are noncommuting
elements of A. Let G = A ∗ B. By the Kurosh subgroup theorem, if H and K are
nontrivial subgroups of G which commute elementwise and hH, Ki is not cyclic,
then they must lie in a conjugate of either A or B. Since A is nondiscriminating
there is a finite set of nontrivial elements {(x1 , y1 ), . . . , (xm , ym )} in A × A such
that no homomorphism of A × A into A fails to annihilate all of these elements.
Similarly there is a finite set of nontrivial elements {(w1 , z1 ), . . . , (wn , zn )} of
B × B such that there is no homomorphism of B × B into B which fails to
annihilate all of these elements. Consider the set of all these elements as well as
(a1 , 1), (a2 , 1), ([a2 , a1 ], 1), and (1, b1 ) as coming out of G × G, where b1 ∈ G\{1}.
Any homomorphism, ϕ, into G which does not annihilate any of the elements in
this set must map G × G into a conjugate of either A or B. To see that, note that
we have guaranteed that niether the image of G × 1 nor the image of 1 × G is
trivial. Moreover, since the image of the commutator is also nontrival the image
of G × G = hϕ(G × 1), ϕ(1 × G)i is not cyclic. The fact that ϕ(G × G) lies
in a conjugate of a free factor now follows from the intial observation using the
Kurosh subgroup theorem. This would then produce a map from the square of
the factor to the factor which did not annihilate any of the given elements in that
factor – contradicting the choice of elements. Hence, G = A ∗ B is, as claimed,
nondiscriminating. It was proven in [BG] under additional hypotheses which were removed by R.
Swan in an addendum to the paper that if F/R is a presentation of a torsion free
group G and R0 is the commutator subgroup of R, then the centralizer of every
element of F/R0 not in R/R0 is infinite cyclic whereas the centralizer in F/R0 of
any nontrivial element of R/R0 is R/R0 .
Example 13. Thus, if R is a proper normal subgroup of a free group F and F/R
is torsion free, then F/R0 is a nonabelian CT group. Hence, such F/R0 are nondiscriminating. It follows, in particular, without consideration of the cardinality of a
generating set, that every nonabelian group free in the variety Am of groups solvable of length at most m(m > 1) is nondiscriminating. More generally, Baumslag
proves in [BFGS] that F/γc (R) is nondiscriminating whenever F is a free group,
R is a proper normal subgroup, F/R is torsion free, c > 1 and γc (R) is the c-th
term of the lower central series of R. (γ1 (R) = R, γn+1 (R) = [γn (R), R].) From
20
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
this one deduces, without consideration of the cardinality of a generating set, that
every nonabelian group free in the variety of all polynilpotent groups of fixed but
arbitrary type (c1 , . . . , ck ) is nondiscriminating. The question of whether or not
there exists a nonabelian, discriminating, relatively free group remains open.
6. Squarelike Groups
There are various equivalents to the property of being squarelike.
Proposition 6.1. ([FGMS2]) Let G be a group. The following three conditions
are pairwise equivalent.
(1)The group G is squarelike.
(2)The ucl(G) = qvar(G).
(3) There is a discriminating group HG such that G ≡∀ HG .
From this one can deduce that squarelike groups G are actually “powerlike” in
the sense that if n ≥ 2 is an integer then Gn ≡∀ G. Indeed, if G is any squarelike
group and I is any nonempty set, then GI ∈ qvar(G) = ucl(G) since quasivarieties
are closed under unrestricted direct products. Then GI is a model of Th∀ (G); so,
every universal sentence of L true in G is also true in GI . On the other hand, G
embeds in GI ; so, every universal sentence of L true in GI must also be true in
G. It follows that GI ≡∀ G.
The authors have proven in [FGMS2] that the class of squarelike groups is
axiomatic using a characterization of axiomatic model classes in terms of closure
properties. Here we exhibit an explicit axiom schema for this class which we
established in [FGS2]. To each ordered pair (w,u) of finite tuples of words on a
fixed but arbitrary finite set {x1 , . . . , xn } of distinct variables and their formal
inverses we assign the following sentence σ(w,u) of L:
∀x(
^
(wi (x) = 1) →
i
_
j
(uj (x) = 1)) →
_
∀x((
j
^
wi (x) = 1) → (uj (x) = 1)).
i
The contrapositive of σ(w,u) is (up to logical equivalence) the sentence τ (w,u)
asserting
^
^
^
^
∃x( (wi (x) = 1) ∧ (uj (x) 6= 1)) → ∃x( (wi (x) = 1) ∧ (uj (x) 6= 1)).
j
i
i
j
Theorem 6.2. (Huber-Dyson [H2] -see [FGS2]) The class of squarelike groups is
the model class of the group axioms and the sentences σ(w, u). Hence, the class
of squarelike groups is axiomatic.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
21
There is an older (and inequivalent) notion of discriminating group due to G.
Baumslag, B.H. Neumann, H. Neumann and P.M. Neumann [Ne]. Let V be a
variety of groups and let the group G lie in V. The group G discriminates the
variety V provided, given finitely many laws uj (x) = 1, none of which holds for all
groups in V, there is a tuple g from G such that uj (g) 6= 1 for all j. The group G
is discriminating in the older sense of Baumslag and the Neumanns just in case it
discriminates the variety it generates. Every group which is discriminating in our
sense is also discriminating in the older sense of Baumslag and the Neumanns,
but not conversely.
Let G be a group lying in a quasivariety Q. We say that G q-discriminates Q
provided that, given finitely many quasi-identities ∧i (wi (x) = 1) → (uj (x) = 1)
, all of which have the same antecedent ∧i (wi (x) = 1) and none of which holds
for all groups in Q, there is a tuple g from G such that wi (g) = 1 for all i and
uj (g) 6= 1 for all j. Call G q-discriminating provided it q-discriminates the
quasivariety it generates. Theorem 6.2 may be viewed as asserting that a group
is squarelike if and only if it is q-discriminating.
In [FGS1] and [FGS2] the authors show that if G is squarelike there exists a
discriminating group HG such that G ≡ HG . Since every axiomatic class must be
closed under elementary equivalence, the class of squarelike groups must be the
least axiomatic class containing the discriminating groups.
Proposition 6.3. ([FGS1], [FGS2]) The class of squarelike groups is the least
axiomatic class containing the discriminating groups.
We now turn to the existence of a finitely generated squarelike group which
is not discriminating.
Example 14. Let H be the subgroup of the group of all permutations of the set
Z of integers generated by the 3-cycle ξ = (012) and the translation η(n) = n + 1
for all n ∈ Z. This group H = hξ, ηi can also be described as the semidirect
product of the group M , of all even parity permutations within the group N of
all permutations of the set Z of integers which move only finitely many integers,
by an infinite cyclic group C = hc; i where the automorphism α(c) : M → M
acts by α(c)(π)(n) = π(n − 1) + 1. (We say that α(c) acts by translation by 1.)
Note that any bijection between N and Z induces an isomorphism between the
infinite alternating group Aω and M . The group H first appeared in print in the
same paper [N] of B.H. Neumann in which the uncountably many nontrivially
discriminating groups Gn of Example 11 in Section 4 were introduced. B.H.
Neumann observed that (independent of n) if K0 is the restricted direct product
22
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
of the family (An(r) )r∈N of alternating groups, then the quotient of G = Gn
modulo K0 is isomorphic to H. We shall prove that H is nondiscriminating.
Subsequently we shall give two different proofs that H is squarelike.
Theorem 6.4. The group H is not discriminating.
We need some preliminary lemmas.
Lemma 6.5. The group H is centerless.
Proof: Since an infinite cyclic group is free, it is projective and the sequence
1 → M → H → H/M → 1 must split. Then H is a semidirect product and
there is a retraction H → hηi. Every element of H is uniquely of the form aη m
where a ∈ M and m ∈ Z. Given a nontrivial element a ∈ M ∼
= Aω there
is clearly an element b ∈ M which does not commute with it since Aω is a
nonabelian simple group; hence, it is centerless. So a nontrivial central element
in H must be of the form aη m with m 6= 0. We cannot have a = 1 since
η m ξη −m = (m m + 1 m + 2) 6= (012) = ξ. Thus, a nontrivial central element in
H must be of the form aη m with a ∈ M \{1} and m 6= 0. But, if any aη m were
central in H, then both aη m and η would lie in CH (η m ) so a ∈ CH (η m ). Thus,
a commutes with η m and η −m . Then η |m| aη −|m| = a. But that is absurd since
conjugation by η |m| moves every integer in the disjoint cycle decomposition of a
by |m| units. So if n is the largest integer moved by a it is replaced by n + |m| a contradiction. Recall that a group G is subdirectly irreducible or monolithic provided
it has a unique minimum normal subgroup M 6= 1. Following Hanna Neumann
[Ne] we shall call the unique minimum normal subgroup of a monolithic group its
monolith
Lemma 6.6. The group H is monolithic with monolith M.
Proof: We must show every 1 N E H contains M .
Case I: For any nontrivial normal subgroup N of H, M ∩ N 6= 1.
Let a ∈ M ∩ N, a 6= 1. Then haiM ≤ haiH ≤ N . But haiM = M since
M∼
= Aω is simple.
Case II: For any nontrivial normal subgroup N of H, M ∩ N = 1.
We shall prove this case cannot occur. If M ∩ N = 1, then M N < H and
MN ∼
= M × N . We have N ∼
= M N/M ≤ H/M is infinite cyclic since N 6= 1.
m
Suppose aη generates N with a ∈ M . We first claim a = 1. For suppose a 6= 1.
Since aη m ∈ CH (M ) we would have aη m ∈ CH (a) and η m ∈ CH (a) would follow.
But, as we saw before, for a 6= 1, this is absurd. Thus, N = hη m i with m 6= 0. But
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
23
then η m would be a nontrivial central element in H = hM, ηi. The contradiction
shows that Case II cannot occur so M is the monolith of H. Lemma 6.7. Every nontrivial normal subgroup 1
M × M nontrivially.
N E H × H must intersect
Proof: Assume 1 N E H × H. Suppose not to deduce a contradiction. Let
pi : H × H → H be the projection onto the i-th coordinate, i = 1, 2. We claim
that M ∩ pi (N ) = 1 for i = 1, 2. To see that suppose that (a1 , b2 η m(2) ) ∈ N where
a1 ∈ M \{1} and m(2) 6= 0. Then hb2 η m(2) iH ≥ M so a1 ∈ hb2 η m(2) iH and there
are h1 , . . . , hk ; ε(1), . . . , ε(k) such that
m(2) ε(1)
m(2) ε(k)
a1 = h−1
) h1 · · · h−1
) hk .
1 (b2 η
k (b2 η
So that
(1, h1 )−1 (a1 , b2 η m(2) )ε(1) (1, h1 ) · · · (1, hk )−1 (a1 , b2 η m(2) )ε(k) (1, hk )
ε(1)+···+ε(k)
= (a1
, a1 )
lies in N and also M × M . Thus, M ∩ p1 (N ) 6= 1 contradicts our assumption that
N ∩ (M × M ) is trivial. Similarly, if (b1 η m(1) , a2 ) ∈ N with a2 ∈ M \{1} we get a
contradiction to the triviality of N ∩ (M × M ). Thus, M ∩ pi (N ) = 1 for i = 1, 2
as claimed. From this we see that pi (N ) = 1 for i = 1, 2 since M is the monolith
of H. Then N = 1 – contrary to hypothesis. This shows that N must intersect
M × M nontrivially. Lemma 6.8. Every nontrivial normal subgroup 1
either M × 1 or 1 × M .
N E H × H must contain
Proof: Suppose (a1 , a2 ) is a nontrivial element of N ∩ (M × M ). If a2 = 1,
then M × 1 = h(a1 , 1)iM ×1 ≤ h(a1 , 1)iH×H ≤ N . Similarly, if a1 = 1, then
1 × M ≤ N . So we may assume that both a1 and a2 are nontrivial. Suppose that
the order of a2 is m. Since ha1 iH = M we can find b ∈ M ∼
= Aω of order n > 1
prime to m. Now, proceeding in a similar manner as before,
ε(1)
b = h−1
1 a1
ε(k)
h1 . . . h−1
k a1
hk .
Then
ε(1)+···+ε(k)
(b, a2
) = (h1 , 1)−1 (a1 , a2 )ε(1) (h1 , 1) · · · (hk , 1)−1 (a1 , a2 )ε(k) (hk , 1)
lies in N . Hence, so does
ε(1)+···+ε(k) m
(b, a2
)
= (bm , 1)
and bm 6= 1. Then, arguing as before, M × 1 ≤ N . 24
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
Proof of Theorem 6.4: If H were discriminating, then there would be a
homomorphism ϕ : H × H → H which did not annihilate either of the elements
(ξ, 1) or (1, ξ). Then neither M × 1 nor 1 × M can be contained in the kernel of
ϕ. It follows that the kernel of ϕ is trivial and ϕ is an embedding of
H × H ,→ H.
That is impossible by Lemma 4.1. The contradiction shows that H is not discriminating. Theorem 6.9. H is squarelike.
First Proof of Theorem 6.9: Let us consider a group G = Gn in the family
of nontrivially discriminating groups of B.H. Neumann discussed in Example 11
of Section 4. Thus G ≤ Πr∈N An(r) ≤ AN
ω . Let K0 be, as before, the restricted
direct product of the family (An(r) )r∈N and let K be the restricted direct power of
N
∼
the family (Aω )r∈N . Under the epimorphism AN
ω → Aω /K, G maps to GK/K =
N
∼
G/(G ∩ K) = G/K0 = H. Moreover, *Aω = Aω /K is the reduced power of Aω
modulo the filter D of cofinite subsets of N. Let d : Aω →*Aω be the canonical
embedding. We claim that Aω and *Aω are universally equivalent with respect
to L. Since Aω embeds in *Aω , every universal sentence true in *Aω must also
be true in Aω . Suppose that that u is a universal sentence true in Aω . Since Aω
is squarelike, it follows from Proposition 6.1 that u is a consequence of a set Q of
quasi-identities true in Aω . But Q is preserved in the reduced product *Aω so its
logical consequence u must also hold in *Aω . Thus, Aω and *Aω satisfy precisely
the same universal sentences of L and are indeed universally equivalent.
Since Aω ∼
= M ≤ H, every universal sentence true in H must also be true in
Aω . Suppose u is a universal sentence true in Aω . Then u is also true in *Aω .
But, since H is embedded in *Aω , u must also be true in H. Thus, Aω and
H satisfy precisely the same universal sentences and are universally equivalent.
Recalling that Aω is discriminating, it follows, again by Proposition 6.1, that H
is squarelike. Let N be the group of all permutations of the set Z of integers which move only
finitely many integers (so that N ∼
= Sω ). Let S be the semidirect product of N
by an infinite cyclic group C = hc; i where the automorphism α(c) is translation
by 1. Clearly, H ≤ S. In [H1] S made an appearance where it is described
as the subgroup of the group of all permutations of the integers generated by a
transposition and the successor function.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
25
Second Proof of Theorem 6.9: Since H ≤ S every universal sentence true
in S must also be true in H. On the other hand, an embedding Sω ,→ Aω will
clearly induce an embedding S ,→ H. Thus, every universal sentence true in
H is also true in S and we see that H and S are universally equivalent. Now
Theorem 2 of [H1] asserts that a universal sentence in the language of group
theory, i.e., our language L, is valid in all finite groups if and only if it holds in S.
Thus, Th∀ (S) = Th∀ (F) = Th∀ (Sω ) where F is the class of finite groups. Then
Th∀ (H) = Th∀ (Sω ); so, H is universally equivalent to a discriminating group and
is therefore squarelike by Proposition 6.1. Acknowledgement
The authors have benefited from communications with the following individuals:
Gilbert Baumslag, Verena Huber-Dyson and Peter M. Neumann. We thank them
for their contributions. The authors would also like to thank the anonymous
referee for many suggestions which improved the style and presentation of the
paper.
7. References
[B] G. Baumslag, Lecture Notes on Nilpotent Groups, Amer. Math. Soc., Providence, 1969.
[BFGS] G. Baumslag, B. Fine, A.M. Gaglione and D. Spellman, “Reflections on
discriminating and square embeddable groups,” Preprint.
[BG] G. Baumslag and K.W. Gruenberg,“Some reflections on cohomological dimension and freeness,” J. Algebra 6(1967), 394-409.
[BMR1] G. Baumslag, A.G. Myasnikov and V.N. Remeslennikov,“Algebraic geometry over groups.I. Algebraic sets and ideal theory,” J. Algebra 219(1999),
16-79.
[BMR2] G. Baumslag, A.G. Myasnikov and V.N. Remeslennikov,“Discriminating
and codiscriminating groups,” J. Group Theory 3(2000), 467-479.
[C] P.M. Cohn, Universal Algebra, Harper and Row, New York, 1965.
[CFP] J.W. Cannon, W.J. Floyd and W.R. Parry,“Introductory notes on Richard
Thompson’s groups,” Ensign. Math. (2) 42(1996), 215-256.
[FGMS1] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, “Discriminating groups,” J. Group Theory
4(2001), 463-474.
26
BENJAMIN FINE ANTHONY M. GAGLIONE DENNIS SPELLMAN
[FGMS2] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman,“Groups
whose universal theory is axiomatizable by quasi-identities,” J. Group Theory
5(2002), 356-381.
[FGMS3] B. Fine, A.M. Gaglione, A.G. Myasnikov and D. Spellman, “The elementary theory of groups,” Proc. Groups St. Andrews 2001 in Oxford, LMS
Lecture Notes 304, Cambridge Univ. Press, 2003, 197-231.
[FGS1] B. Fine, A.M. Gaglione and D. Spellman,“The axiomatic closure of the
class of discriminating groups,” Preprint.
[FGS2] B. Fine, A.M. Gaglione and D. Spellman,“Discriminating and squarelike
groups I: Axiomatics,” To appear in Contemporary Mathematics, AMS.
[G] G. Gratzer, Universal Algebra, Van Nostrand, Princeton, 1968.
[Gr] R.I. Grigorchuk, “On the growth degrees of p-groups and torsion-free groups,”
Math. Sb. 126(1985),
194-214 (English translation: Math. USSR Sbnornik 54(1986), 185-205).
[GS] N. Gupta and S. Sidki, “On the Burnside problem for periodic groups” Math.
Z. 182(1983), 385-388.
[H1] V. Huber-Dyson, “A reduction of the open sentence problem for finite groups,”
Bull. London Math. Soc. 13(1981), 331-338.
[H2] V. Huber-Dyson, Private communication.
[Hi1] G. Higman, “Subgroups of finitely presented groups,” Proc. Royal Soc.
London Ser. A(1961), 455-475.
[Hi2] G. Higman, Finitely Presented Infinite Simple Groups, Notes on Pure Math.
8, I.A.S., Austral. Nat. Univ., Canberra, 1974.
[HM] R. Hirshon and D. Meier, “Groups with a quotient that has the original
group as a direct factor,” Bull. Austral. Math. Soc. 45(1992), 513-520.
[J] J.M. Tyrer Jones, “Direct products and the Hopf property,” J. Austral. Math.
Soc. 17(1974), 174-196.
[Ka] M. Kassabov, “On discriminating solvable groups,” Preprint.
[KhM] O. Kharlampovich and A.G. Myasnikov, “Tarski’s problem about the elementary theory of free groups has a positive solution,” Electron. Res. Ann. AMS
(1998), 101-108.
[K] A.G. Kurosh, The Theory of Groups, Vols. I and II, Chelsea, New York, 1956.
[MS] A.G. Myasnikov and P. Shumyatsky, “Discriminating groups and
c-dimension,” To appear in J. Group Theory.
[N] B.H. Neumann, “Some remarks on infinite groups,” J. London Math. Soc.
12(1937), 120-127.
[Ne] H. Neumann, Varieties of Groups, Springer, New York, 1967.
DISCRIMINATING AND SQUARELIKE GROUPS II: EXAMPLES
27
[Se1] Z. Sela, “Diophantine geometry over groups VI: the elementary theory of a
free group.” Preprint.
[Se2] Z. Sela, “Diophantine geometry over groups VIII: the elementary theory of
hyperbolic groups.” Preprint.
[S] W. Szmielew, “Elementary properties of Abelian groups,” Fund. Math. 41(1955),
203-271.
Prof. Fine.
Department of Mathematics
Fairfield University
Fairfield, CT 06430
Prof. Gaglione
Department of Mathematics
U.S. Naval Academy
Annapolis, MD 21402
amg@usna.edu
Prof. Spellman
Department of Mathematics
Temple University
Philadelphia, PA 19122