I ntrn. J. Mah. Math. Sci.
Vol. 3 No. 4 (1980) 719-730
719
EXTENSIONS OF GROUP RETRACTIONS
RICHARD D. BYRD, JUSTIN T. LLOYD
ROBERTO A. MENA, AND J. ROGER TELLER
Department of Mathematics
University of Houston
77004 U.S.A.
Houston, Texas
(Received June 18, 1979 and in revised form February 29, 1980)
ABSTRACT.
In this paper a condition, which is necessary and sufficient, is
determined when a retraction of a subgroup
extended to a retraction of
G.
H
of a torsion-free group
G
can be
It is also shown that each retraction of a
torslon-free abelian group can be uniquely extended to a retraction of its divisible closure.
KEY WORDS AND PHRASES. Group retraction, extending group rrations, torsionfr--ee abelian group, divisible closure..
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
Secondary, 20K15, 20K20.
i.
Primary, 06A75, 20E99
INTRODUCTION.
The concept of a retractable group was introduced in [ 2] and there it was
shown that the class of lattice-ordered groups is a proper subclass of the class
of retractable groups.
It was shown in [4
that group retractions induce "non-
standard" automorphisms of the semigroup of finite complexes of a torsion-free
720
R. D. BYRD, J. T. LLOYD, R.A. MENA AND J.R. TELLER
In this paper we consider the following question:
abelian group.
torison-free abelian group,
H, then when can
H
is a subgroup of
G, and
be extended to a retraction of
T
G?
T
G
if
is a
is a retraction of
In Theorem 5.7 we give
a necessary and sufficient condition for the existence of such an extension.
key to proving this theorem is Theorem 3.2 where it is shown that if
normal subgroup of a group
is a retraction of
set
G.
A
such that
g
-i
(AT)g, then
is a
G/H can be linearly ordered and
such that for each
(g-lAg) T
H,
of
H
G
H
g
T
G
The
T
and each finite nonempty sub-
can be extended to a retraction of
This theorem is a generalization of a corresponding theorem in the theory of
lattlce-ordered groups (see [1],[7], or [I0]).
If
G
is a torsion-free abelian group and
D
is a divisible closure of
then, in Theorem 4.7, we show that each retraction of
to a retraction of
D.
G
G,
has a unique extension
Again this theorem generalizes a well-known result in the
theory of lattlce-ordered groups.
The key in the proof of Theorem 4.7 is Corol-
lary 4.4 where it is shown that each retraction of an abellan group satisfies
condition
(6).
(Definitions will be given in Section 2.)
obtain a partial converse of Theorem 4.7.
In Corollary 4.8, we
An immediate consequence of Corollary
5.8 is that each torsion-free abellan group admits an infinite number or retractions.
2.
PRELIMINARIES.
In this section we give some definitions and results from [2] that will be
used in this paper.
Throughout this paper,
plicatively and with identity
F(G)
will denote the collection of all
Then
F(G)
is a join monoid, that is,
G.
a join semilattice in which
A v B
{abla
homomorphism
A
@
and
of
b
F(G)
be called a retraction of
A u B,
B}, A(B v C)
into
G.
will denote a group,written multi-
I, and
finite, nonempty subsets of
AB
G
G
F(G)
is a monoid in which
AB v AC, and
such that
{g}
F(G)
(A v B)C
g
AC v BC.
for every
g
G
A
will
We will denote by Ret G the collection of all
is
EXTENSIONS OF GROUP RETRACTIONS
If Ret G is nonempty, then
G.
retractions of
G
721
is said to be a retractable
The class of retractable groups is a proper subclass of the class of
group.
torslon-free groups [2, Theorem 2.2 and Example 2.7].
G
If
A
E
F(G), then o
Ret G [2,Theorem 2.1] and
E
induced by the lattice-ordering of
Let
o
is a lattice-ordered group and
o
Ret G.
F(G)
G.
of
G
orderings of
Ker o
If
F(G), then
semilattice) of
traction)
}.
o
o
G
o, denoted by Ker o,
is the set
(resp., convex sub-
is a convex subset
-re-
(resp.,
is said to be a convex retraction
There is a one-to-one correspondence between the latticeand the
A subgroup
F(H).
A
every
for each
is called the retraction of
-retractions of
G
H
G
of
In [2, Theorem
[2, Corollary 3.3].
3.2] six conditions are given, each of which is equivalent to
traction.
v A
Ao
G.
Then the kernel of
AO
and
is given by
is said to be a
o
being an
Ao
G-subgroup if
H
-re-
for
In this paper retractions that satisfy the following condition
are of prime importance:
(6)
n
if
o
E
{gl
Ret G,
gm }
F(G)
and
is a natural number, then
{g’’’’’gmn}O ({gl’’’’’gm }O)n"
If
G
o
is a lattice-ordered group and
ordering of
o
G, then
satisfies
(6)
is the retraction induced by the lattice-
if and only if the lattice-ordering is
(This can be proven by [5, Theorem 1.8].)
representable.
given of groups that admit retractions satisfying
In
(6), but do
[3]
examples were
not admit lattice-
orderings.
Let
O
Ret G
0-G-subgroup_. of G
H(AO)
H({hlg I
4.10] and is a
H
and
if
A
hngn}O).
be a subgroup of
gl
Every
gn }
G.
F(G)
Then
and
0-G-subgroup of
h
H
is said to be a
I,...,hn
H
G
[2, Corollary
is pure
implies that
G-subgroup [2, Theorem 4.2].
The natural numbers will be denoted by
N, the integers by
Z, and the
722
BYRD
R.D.
Q.
rationals by
J.T.
A_c
If
G
LLOD R.A. INA AI J.R. TELLF
and
N, then
n
An= {al...anla
I’
A}
.,a
and
A (n)
3.
{anla
A}.
LEXICOGRAPHIC EXTENSIONS.
In 1942, F. W. Levi [10, p. 260] (or see [I, p. 289] or [7, p. 20]) gave a
necessary and sufficient
conditi.on
that a partial order of a subgroup of a group
may be extended to a partial order of the group.
The main result of this section
gives a sufficient condition for the extension of a retraction of a subgroup to a
retraction of the group.
THEOREM 3.1:
(1)
If
(ll)
h
I
h
H
o
Ret G
is a
H
and
Ret H, then o
T
H
n
Let
The proof of Theorem 3.1 is immediate and will be omltted.
be a subgroup of
extends
0-G-subgroup of
G
if and only if
T
G, then the mapping
o*
THEOREM 3.2:
T
Ret H
G/H
Let
such that for every
g
(G/H,<)
o
that extends
(1)
of
H
G
is a
<
of
H
G
(ill)
T
is an
{gl,...,gn}
Hg
I
<
o*
G/H
o
H
is a normal
o).
A
F(H),
G, and
(g-lAg) T
g
-i
(AT)g.
Then there is a re-
is a convex
G/H
induced
o-subgroup of
is convex;
if and only if
F(G)
<
H
induced by
is the retraction of
G/H; and hence,
is convex if and only if
A
and
be a normal subgroup of
G,
T
Let
and
Moreover,
T.
E-retractlon
e Ker O
,gn}O)
I
and every
(ll)
PROOF:
Ret G
is a linearly ordered group.
0-O-subgroup of
by the linear ordering
H({g
be a group,
Suppose further that
traction
o
(called the retraction of
G
o.
given by
{Hgl,...,Hgn}O*
is a retraction of
er
T c
H.
It was shown in [2, Theorem 4.3 (1)] that if
0-G-subgroup of
Ker
{gl,...,gn }
if and only if
{hlg I ’’"’hngn }O
imply that
G.
o
is an
where
Hgp_ 1 < Hgp
Hg
n.
E-retractlon.
G;
EXTENSIONS OF GROUP RETRACTIONS
p < i < n,
Then for
glgn
We must show that o
-1
gpgn
Since
dgp
Thus
every
Hx
I
n
G.
g
{gpg 1, ...,gngn-1 }T
h
Suppose that
is well-defined.
u
and it follows that
A be as above,
Let
HXq_l < HXq
_<
Lt
H.
E
and define Ao
hg
n.
(gPg]r ,...,gngp-1}T.
d
H,
E
hg
I
723
{Xl’’’’’Xm }
d
x-i
{Xqm
B
HXm,
and
{g}o
Clearly,
is a function.
for
g
F(G), where
-1
XmmX
}T.
BO
Then
dXm,
and if
{gpXq gn
c,
’’’’’gn
(AB)O
CgnXm.
Now,
1
{gn-1 gpXqXm-i
gn cg.
{g
cg
Therefore,
hgnd
n
and so
-i
-1
’gn gnXmXm }’t"
"’’’gn gn }T{Xq xm ’’" .,xmxm-I}T
-i
igp
(AB)O
-i
CgnXm
hgndxm
(Ao)(Bo).
Hence
e RetG.
If
-i
h h
{_h
,h }
I
t.
E
n
Thus
hl {hlhl
Ker T, then
=
Ker T
Ker o
1
,...,hnh- }T
n
and, by Theorem 3 i,
G
and so {
hl ,...,hn
extends
The verification of (i) is immediate from Theorem 3.1 and [2, Theorem 4.3
(ii)], the verification of (ii) is straightforward, and (iii) is well-known from
the theory of lattlce-ordered groups.
COROLLARY 3.3:
subgroup of
G, and
If
G
T E
Ret H, then
is a torsion-free abelian group,
T
H
is a proper pure
can be extended to a retraction of
G
in at least two ways.
PROOF:
If
H
is a pure subgroup of
G, then
G/H
is torslon-free.
torsion-free abelian group can be linearly ordered [5, p. 0.4].
proper subgroup of
Hence
T
G,
G/H admits
Since
H
at least two distinct linear orderlngs.
can be extended in at least two ways to a retraction of
G.
Each
is a
R. D
72
BYRD, J T. LLOYD, R.A. MENA AND J.R. TELLER
4. DIVISIBLE EXTENTIONS.
The main result of this section is that each retraction of a torslon-free
abelfan group can be uniquely extended to a retraction of its divisible closure.
Similar results for partially ordered groups may be found in [8],[6], or [5,
Chap. 4].
If
G
is a lattlce-ordered group,
for all
i
and
j, and
such that
mn
(gl v...vgm)n
N, then
n
G
gl’’’’’gm
glv...vg
n
glgj
gjgl
[5, p. 0.17].
In Theorem 4.3 we show that this identity generalizes to retractable groups.
this end, we state two lemmas.
To
(A proof of Lemma 4.1 can be given by induction
and Lemma 4.1 can be used to prove Lemma 4.2.)
LEMMA 4.1:
gg
for all
G
If
and
i
j, and
An= {glk I "’’ginkm
LEMMA 4.2:
for all
i
If
and
.N,
n
i
for all
i
and
PROOF:
4.2,
(Ao)
If
j, and
Since
An(m+l)
nm({g,
A
< m, then
Ret G, A
N, then
n
k
r
> 0
p
n
N
Anm{gl,...,gmn}
gmn} )O"
Hence,
n}
p >
gig
ggi
(m-1)(n-1),
Ap+n.
,gm }
{gl,...,gm}
F(G)
{g?,...,gmn}o
n
n
}o
{gl,...,g
m
Z k
r
such that
such that
(AO)
such that
(AO)
(AO) n(m+l)
Therefore, (AO) n(m+l)
n
and
F(G)
is a homomorphism,
O
gigj
r=l
{gl,...,gm }
N, then for each
n
O
such that
then
-< r
AP{g
THEOREM 4.3:
F(G)
m
if
G is a group,
j, and
{gl,...,gm }
A
is a group,
n.
n.
gi gJ
ggij
(An(m+l))o, and by Lemma
n
n}) O
(Anm{gl,...,g
m
We state the following three corollaries which will be used numerous
times
in the sequel.
COROLLARY 4.4:
If
G
is an abelian group and
If
G
is an abelian group,
o
Ret G, then
o
satisfies
condition (6).
COROLLARY 4.5:
and
Ret G, then
G(n)
is a
o-subgroup of
n
G.
N,
G(n)
{gn
g
G},
Thus, the largest divisible
725
EXTENSIONS OF GROUP RETRACTIONS
G
subgroup of
o-subgroup.
is a
Let
COROLLARY 4.6:
H,
G
n
H
for some
n
o e Ret G and H
If
G,
o-subgroup of
is a
G.
be a divisible closure of
D
be a torslon-free abelian group and
D
n
whose
Then
{d
n
D,
is a subgroup of
D
DmDn DI. c.m
d e D
If
g e G, we write
n-th power is
g.
For
o
G
If
g I/n
D
onto
e
G}.
D
n D
D
m
n
from
G
into
and
g.c.d. (re,n)’
D
for that unique element in
n
D
gl/n"
g@n
by
n
is an abellan group,
o
Ret G, then
G
n
u D
nN n
D
n e N, define
is an isomorphism of
n
d
and
n D
neN n
G
(re,n)"
THEOREM 4.7:
and
g
N, let
n
Then
G.
in
and
G, and
a subgroup of
H,.
then so is
For
G
g
H
H
be an abellan group,
g
be the pure closure of
Let
G
D
G,
is a divisible closure of
can be uniquely extended to a retraction
of
T
D.
Moreover,
for each
(i)
D
i/n
T-subgroup of
gm }In
N
and
D;
{gl,...,gm }
Ker }
if and only if
Ker T
and
F(G) s Ker T;
Ker
Ker
(ill)
is a convex subset of
Ker o
(iv)
T
PROOF:
A
is, for
s
an
F(D); hence o
E-retraction of
For each
e
n
Ret D
n
Let
T
is an
if and only if
-retraction of
F
Ker
n
n
({gl@ I
u
n
nN
is
T
if and
-ioe
n
n
where
n
is given above.
F(Dn)’
AO
c
is a convex
D.
c
N, let
n
’gm }
{gl
F(G)
is a convex subsemilattice of
a convex subsemilattice of
only if
F(G)
F(D)
subset of
Then
is a
n
i/n
{{gl
Ker
(ii)
N,
n
’gmn-l}c) n
The verification that
T
is the unique
That
R. D. BYRD, J. T. LLOYD, R.A. MENA AND J R
726
extension of
D, and the remainde of the theorem is rou-
to a retraction of
O
TELLER
tine.
COROLLARY 4.8:
group of
G.
G, then
G/H
sure of
H).
PROOF:
H,
Ret H
T
H
denotes the pure closure of
If
G
Hence,
G.
H,
T
in
5.
T
G.
to a retraction of
H,
If
o-
is a
H,.
extends uniquely to a retraction of
Thus, by
G.
is a divisible closure of
D
G,
to a retraction of D, then it
o
is the unique extension of
T
is easy to show that the collection of
phic to the
has a unique extension to
T
G, then by Corollary 4.6,
is a torsion-free abelian group,
Ret G, and
be a sub-
is contained in a divisible clo-
G
o be the unique extension of
Let
Corollary 3.3,
such that
(or equivalently,
is torsion
H
be a torsion-free abelian group and
G
If there exists
subgroup of
o
Let
0-o-subgroups of
G
is lattice isomor-
D.
0-T-subgroups of
EXTENDING RETRACTIONS.
In this section we prove the main result (Theorem 5.7) of this paper, namely,
if
is a torsion-free ableian group and
G
H
a necessary and sufficient condition that a retraction of
Let
THEOREM 5.1:
be a subgroup of
D
G
K
can be extended to
G, and
o
o
G,
a divisible closure of
Ret G.
if and only if for each
Moreover, if
G.
D
be an abelian group,
that contains
tended to a retraction of
o-subgroup of
H
G.
a retraction of
K
G, then we give
is a subgroup of
n
o can be ex(n) n G is a
K
Then
N,
K, then it is
extends to a retraction of
unique.
PROOF:
O
of
Let
T
can be extended to a retraction
9
to a retraction of
-subgroup of
K
to a retraction of
be the unique extension of
D.
and hence, a
If
of
n
K, then
T
D.
If
is the unique extension
N, then by Corollary 4.5,
T-subgroup of
D.
Therefore,
K
K
(n)
(n) n G
is a
is a
727
EXTENSIONS OF GROUP RETRACTIONS
D.
Y-subgroup of
IF(G)
Since
K(n)
it follows that
0
G
is a
G
for each
u-subgroup
G.
of
K
Conversely, suppose that
{h
Let
F(K)
,hm
,hmn}
I
{h n
I
that
F(G).
Now,
K(n)
n
n
n G
K
n
Then there exists
a.
G
D
Since
and hence,
{
and
o
such
n
m
hi’’" hn}o.
n
{hi,..., hmn} F(K(n)
h n}.
m
N
n
N.
n
(6),
satisfies
{h
n G.
o-subgroup of
is a
hm }T
Since
(n)
D
T-subgroup of
is a
n G
G-subgroup of
is a
hn}o
m
=-hi’’"
(n)
{h I,
and
a
a
o,
K.
a
is torsion-free,
n G).
Hence,
K
Therefore,
K.
can be extended to a retraction of
The uniqueness is immediate from Theorem 4.7.
From the proof of Theorem 5.1, we have
Let
COROLLARY 5.2:
G,
G
D
that contains
G
and
be a subgroup of
K
K(n)
such that
c
K(n)
unique extension to a retraction of
be a divisible closure of
D
be an abel+/-an group,
G, and
Ret G.
O e
u-subgroup of
is a
If there exists
o
G, then
has a
K.
In the following considerations (up to Corollary 5.6), for obvious reasons,
we use additive notation for the binary operations on groups.
Q
group of
H
and define
easily verified that
H
+/-
+/-
{rlr
e Q
rh
and
is a subring of
Q
H
containing
from the theory of abel+/-an groups of rank i that if
taining
H
and
H
+ {0},
then
In [2, Example 5.7]
{Or[r
Ret Q
all
and
e
Q},
H
i
is a subring of
all of the retractions of
where
Or
is defined by
for every
AOr
H
Let
be a sub-
H}.
h
It is
z. Moreover,
it follows
Q
is a subgroup of
K
con-
K+/-.
were exhibited, namely,
Q
(r + l)max A
r min A, for
A e F(Q).
COROLLARY 5 3:
If
H
is a nonzero subgroup of
COROLLARY 5.4:
If
H
and
o
Ret H, then
o
K
are subgroups of
{o
Ret H
Q
then
Q
such that
can be uniquely extended to a retraction of
H
r
r
! K, H
K.
The preceding corollary is not true for abel+/-an groups of rank 2, as we
+/-
H }
+ {0},
728
R.D. BYRD, J.T. LLOYD, R.A. MENA AND J.R. TELLER
illustrate with the following example.
EXAMPLE 5.5:
Let
Z
K
by the cardinal ordering of
(I,-I)
and
2H n K.
(I,0)
ever,
2H.
K.
Q
the cyclic subgroup of
Z
Q
and let
Let
(1/2,-1/2).
generated by
2H n K
Then
K
<(1/2,-1/2)>
2H
v (i,-i)
2K
induced
denotes
+ <(I,-i)>
(I,0)
K
u-subgroup of
is not a
can not be extended to a retraction of
5.1,
-retractlon of
K + <(,-1/2)>, where
H
{(0,0),(i,-i)}-- (0,0)
Now
Thus,
be the
K.
How-
and so by Theorem
H.
The next corollary is an immediate consequence of Corollary 5.2.
COROLLARY 5.6:
(n)
Ret G
G
If
then
is a torslon-free abelian group,
N, and
n
has a unique extension to a retraction on
G
We next turn our attention to the problem of extending a retraction of a
H
subgroup
of a torison-free abelian group
THEOREM 5.7:
and
Ret H.
T
Let
G
G
be a torsion-free abellan group,
H
a subgroup of
Then a necessary and sufficient condition that
tended to a retraction of
G
is that
G
(n)
G.
to a retraction of
n H
is a
G,
can be ex-
T
T-subgroup of
H
for each
n N.
Suppose that
PROOF:
G(n)
s H
is a
T-subgroup of
We first show that there is a pure subgroup
T
can be extended to a retraction of
H,
that contains
for each
(MI,9 I)
N}.
n
<
(M2,92)
9
IF(H)
Ret M,
(
Then
(H,T)
closure of
of
{c
and
p
C.
N
I
Let
G.
M
I c__ 2
(M,9) of
if and only if
M
in
Let
X
c
m}
such that
D
D.
.
T, and
M
G
G(n)
and
M
9
and let
m
c
G
np)
o M.
Since
such that
-subgroup of
M
define
By Zorn’s Lemma,
.
M
is a pure sub-
C
be the divisible
can be extended to a retraction
C). Then, we assert that
(n)
F(G
n C), then G (n) a (G n C)
c p}
N.
G
21F(MI) 91"
G
H
n
is a subgroup of
is a
We show that
IF( G
{cp,-
containing
(MI,91),(M2,9 2)
be a divisible closure of
Then, by Theorem 4.7,
for each
= ,
of
{(M, vIM
Let
and for
there exists a maximal element
group of
M.
M
H
(G C,X)
(n)
G
If
n
N
C, and there exists
satslfles
()
and
729
EXTENSIONS OF GROUP RETRACTIONS
G(nP)
Hence,
M.
{Cl,...,Cm} X {Cl,...,Cm}
i, we have shown that
n
G
0
(n)
C
0
3.3,
X-subgroup of
is a
(M,9),
mality of
X
G
0
M.
C
Ret(G
G n C.
a retraction of
(n)
C), and for
0
n
n C.
In particular, when
arbitrary
Therefore, (G n C,X)
Consequently,
can be extended to a retraction of
9
G
and by the maxi-
G
and hence,
By Corollary
G.
is pure in
M
(n > 0),
T
can be extended to
G.
The converse is immediate by Corollary 4.5.
By Corollary 5 3,
H
group
that if
of
G
Z
and each
r
{
r
G
+/-
Ir
{Orlr
Z }
Ret Z,
is a torsion-free abelian
the cyclic subgroup of
G.
Ret Z
H
g
Thus, for each sub-
u-subgroup of
is a
group and
generated by
Z}.
It follows
Z.
G, then each retraction of
g
can be extended to a retraction of
(Hence, each torsion-free abelian group has at least a countably infinite
number of retractions.)
Combining Corollary 5.4 and Theorem 5.7, we have a
stronger result than this.
COROLLARY 5.8:
cyclic subgroup of
traction of
Let
G.
G
be a torsion-free abelian group and
Then every retraction of
H
H
be a locally
can be extended to a re-
G.
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2.
R. D. Byrd, J. T. Lloyd, R. A. Mena, and J. R. Teller, Retractable groups,
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