I ntcrn. J. Math. Math.
Vol. 2 (1979) 69- 80
69
THE GROUP OF HOMOMORPHISMS OF ABELIAN TORSION GROUPS
M.W. LEGG
Department of Mathematics
Moorhead State University
Moorhead, Minnesota 56560
U.S.A.
(Received February I0, 1978)
ABSTRACT.
Let G and A be abelian torsion groups.
complete set of invarlants for Hom(G,A).
In[bq., R. S. Pierce develops a
To compute these invariants he introduces,
and uses extensively, the group of small homomorphisms of G into A.
some of Pierce’s methods, Fuchs characterizes this group in
this paper is to characterize
Hom(G,A)
Also, using
[I]. Our purpose
in
in what seems to be a more natural manner
than either of the treatments just mentioned.
KEY WORDS AND PHRASES. Abzn Torsion Groups, Group of Homomorphsm, and Ulm
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
1
20KI0, 20K30.
INTRODUCTION.
It is well-known (Harrision [2]) that Hom(G,A) is an algebraically compact
group.
In [3] Legg and Walker use elementary techniques to show that algebraically
compact groups are characterized by their Ulm invariants and their torsion free
M.W. LEGG
70
In this paper we will make use of methods introduced in [3] to cal-
numbers.
culate these invariants for Hom(G,A).
The notation used here will mainly conform to that of Fuchs [I].
paper group means additive abelian group.
The maximal divisible subgroup of the
0, G is called reduced.
If dG
group G will be denoted dG
In this
We will use G
p
to
denote the p-primary component of G and tG to represent the torsion subgroup of G.
If Q is the group of rationals and Z is the group of integers, then
is
Z(n) denotes the cyclic group of order n.
denoted Z(p).
2.
(Q/Z)p
PRELIMINARIES.
A group G is called algebraically compact (Maranda 4) if for every group X,
any homomorphlsm from a pure subgroup of X to G can be extended to a homomorphlsm
of X to G.
We now recall two definitions and the main result from the paper by Legg and
Walker that was mentioned above.
Let G (n) denote
Let G be a group, p a prime, and n a non-negatlve integer.
the dimension of
That is,
Gp(n)
(pnG)Ep/(pn+IG)p
P
as a vector space over the integers modulo p.
is the n-th Ulm invarlant of G (with respect to
If G is a group and p is a prime, then
G0( p
p).
shall denote the dimension of
G/(pG+tG) as a vector space over the inte{ers modulo p.
We call
GO( p
the p-th
torsion free number of G.
The Ulm invariants and the torsion free numbers are a complete set of invariants
for algebraically compact
.groups.
-
That is, two algebraically compact groups A and
B are isomorphic if and only if
(i)
dA
dB
(ii)
A0(p)
B 0(p) for all primes p, and
(iii)
Ap(n)
Bp(n)
integers n.
for all primes p and all non-negative
71
HOMOMORPHISMS OF ABELIAN TORSION GROUPS
We now proceed to calculate Horn (G,A)(n) and
P
Hom0(G,A)(p)
for the algebrlcally
compact Hom(G,A).
It will suffice, as we shall show later, to confine our calculations to p-
So, for the time being, all groups will be understood to be of
primary groups.
this type.
Our calculation of the Ulm invariants of Hom(G,A) requires that we determine
the adjusted component of Hom (G,A).
3.
This result is interesting in itself.
HOMG A).
THE ADJUSTED COMPONENT OF
The following definition and results are due to Harrison [2].
These results
were subsequently obtained by Legg and Walker [3] using quite elementary methods.
A reduced algebraically compact group is called adjusted if it has no torsion free
direct summands.
A reduced algebraically compact group A is the direct sum of an
adjusted algebraically compact group and a torson free algebraically compact group.
The adjusted component of A is unique.
and only if G/tA
d(A/tA).
Further, G is the adjusted part of A if
(For an arbitrary G, the adjusted component of the
reduced part of G is called the adjusted component of G).
The adjusted component of Hom(G,A) will be denoted AdJ(G,A) and its elements
will be called adjusted homomorphisms.
THEOREM 3.1.
AdJ(G,A)
m > 0 there is an integer n
(il)
(pnG)[pm]
PROOF.
c__ Ker
.
is the set of all e
Hom(G,A) such that for each integer
>- 0 such that (i) (pnG)
Let S(G,A) be the
subgup
c__ p
A and
of Hom(G,A) consisting’ f those homomorphisms
that satisfy conditions (i) and (ii) of the theorem.
Let
e AdJ(G,A) and m >_ 0 be
an integer.
Since Adj
(G,A)/tHom(G,A)
an integer n > 0 so that
d(Hom(G,A)/tHom(G,A)), there
pn(e- pms)
0.
is a
8 e Adj (G,A) and
It follows that Adj (G,A)
c__ S(G,A).
By the remarks preceding this theorem, we will be done if we show that
M.W. LEGG
72
S(G,A)/tHom(G,A)
is divisible.
Hom(G,A) is q-divislble for primes q # p and since Hom(G,A)/S(G,A)
Since
clearly q-torsion free, it follows that S(G,A)/tHom(G,A) is q-dlvlsible.
e(pnG) c__
pn+ 1A and
Let B
E
B.i and C
i= I
where B
i
and C
B
write G
Since
(pnG)[p] _c
i
are direct sums of cyclic groups each of order p
r+ Bn+I<+Gn+I
I
n+ 1 A
_c p
e(pnG)
81: pn+l Gn+ 1 An+ I
/
defined.
by 8
n+ i
p
essential in
An+ I
81
A
and
l(pn+Ix)
An+l, we
(pnx).
Let D be the divisible hull of
we can extend
Gn+l,
Ker e
C.i be basic in G and A respectively,
l
i= I
c__
82(Gn+ I)
82(x)
for x e
Gn+ I.
Then
i
+ Cn+ I
I
.
Then we can
An+ I
can define
It is easy to check that
82
Since p
from
Gn+ I
n+l
Gn+ 1
to D.
is well-
81
is essential in
Also p n+l.
an+ I
is
An+ I.
Finally, define the homomorphism 8:G
and 8(x)
C
An+ I.
to a homomorphism
so that
We show
S(G,A) and choose n > 0 such that
Let e
now that it is p-divisible.
is
/
pn(=
A by 8(x)
PS)
0 for x e B
0, since
O Bn+1
I
8(pn+Ix)
s(pnx)
for
xG.
It remains to show that 8 e S(G,A).
pn+ 18
e S(G,A).
(pnG)
pm+n+iA
and
(pn+l G)[ pro]
wished to show.
e S(G,A); thus
(pnG)[ pro+l]
c
Ker py.
implies that y
Hence p
n+l
B
Also
(pn+ix)
Suppose
Choose n > 0 such that
Let m > 0 be an integer.
n+IG) py(pnG) _c pm+n+iA_c pm+nA.
7 (P
n+l
p
x
pn
We must show that Hom(G,A)/S(G,A) is p-torsion free
y e Hom(G,A) and py e S(G,A).
py
It is routine that
Then
(pn+IG)[ pm]
py(pnx)
_c
Ker 7, since
0. Thus y
E
S(G,A), as we
S(G,A) implies that S(G,A)/tHom(G,A)
is divisible.
The proof is complete.
We next investigate the functorial behavior of Adj (G,A).
first lemma depends on the following result [3]:
The proof of our
If A is pure in B, then there
HOMOMORPHISMS OF ABELIAN TORSION GROUPS
C/A divisible and A pure in C.
exists a group C, containing B, with
LEMMA 3.2.
73
If A is pure in B and b
can be extended to an
AdJ(A,X), then
adjusted homomorphism of B into X.
It is immediate that the restriction of an adjusted homomorphism is
PROOF.
Hence, by the remark preceding this lemma, we can assume that B/A
adjusted.
divisible.
m
p
B with o(b)
Let b
(pnA)[pm] c__Ker
Choose n so that
pure in B and B/A is divisible, there exist b
b
Pn + an
and o(b
n)
o(b
n
p
o(bn) --o(bn + A) the fact that
Now define :B / X by (b)
(an).
(pkA)[pm]
Suppose also
O(bk)
o(b
c__ Ker
(ak) so that
n)
Hence _(a
,
and assume k > n.
A so that p a
Thus
n
A[pm]
Hence a
0
n
pk-nbk)
a
K
Thus, since
Write b
pn(bn
We have that
n
A is pure in B, there is an a
pmn
0 implies that
Alpm].
+ A). Then ak
k
pn + A.
+A
b
A is
Since
A such that
B and a
+ A). Note that
.
is
Pk + ak’
an.
a
k
It is routine to check that
is well-deflned.
Since
"-’(pnA)tpmj.
a
n
K
wlth
is
a homomorphism.
Finally, we show that
an integer n > 0 so that
with o(b)
o(b n)
p
o(b
k
and b
p
+ A). Note
n
(pnA)[pm] c__ Ker
that a
by purity, there is an a
(an)
Hence (b)
Now let b
(pSA) [ pk]
c
Ker
.
b
I" Choose
c
A such that p a
say b
Let b
P’I’-pSb s +
n
(pnB)[pm]
with o(b)
p
k
We have shown that
pnx.
as with o (b s)
Thus
Adj (B,X).
(pnB)[pm]
5_ Ker
(pnA)[pm].
n
Choose s > n so that
o (b
n
(a s)
.
It follows that a
a
so that, by purity, there is an a E A such that p a
follows that (b)
Let b e
pn + an and
a so that
We have pn(bl-b n)
A[pm].
n
X.
A so that b
B, an
n
0;i.e. we have shown that
pnB,
(pnA) c__ p
and
A[pk]
Then there is
Let m > 0 be an integer.
is adjusted.
+ A)
s
a ,i.e. a
Then
E
pSbs
pnb I
pnA.
It
(pnB) c__ pm+nx.
Since
extends
,
the lemma is proved.
a
s
M.W. LEGG
74
Hom(A,X), then
Adj (X,Y) and a
If
Thus u determines a homomorphism u*:Adj (X,Y)
THEOREM 3.3.
(*) 0
/
AdJ (B/A,X)
.
Let 0
be pure exact.
*
/
q,
Hence
We must show that
where
:B/A
is adjusted.
adjusted, there is an integer n
>_
Then
0 is split exact for arbitrary X.
X:b
+ A
Let m
>_
/
u.
Adj (A,Y), where *()
/
Clearly q* is a monomorphism and Im q*
PROOF.
A c__Ker
%*A + B n+ B/A + 0
Adj (B,X) i* AdJ (A,X)
AdJ (A,Y).
it is routine to show that
0 such that
=__ Ker
Ker *.
Let
i*.
Then
(b).
0 be an integer.
(pnB)[pm] c_ Ker
is
Since
and
(pnB)c_ pmx.
.
PI
(pn(B/A))[pm], say b + A
+ A with o(b I) o(b I + A). Then
e (pnB)[pm] so that (b+A)
(pl O. Hence (pn(B/A))[pm] =__ Ker
is adjusted, which shows
is adjusted. Hence
Clearly (pn(B/A)) =_ pmX. Thus
Let b
+A
PI
that Ker
*
Im q*.
c
The fact that
*
We have shown exactness at AdJ (B,X).
is onto is a consequence of Lemma 3.2.
By Fuchs [I], n*(Hom(B/A,X)) is pure in Hom(B,X).
Hom(B/A,X) and q*(Adj (B/A,X)) c_ AdJ (B,X),
s-minuend of
is pure in Adj
AdJ(B/A,X)
it follows that
is a
*(AdJ (B/A,X))
(B,X).
Hence (*) is pure exact.
split exact.
Since
Since
AdJ(B/A,X)
is algebraically compact, (*) is
The theorem is proved.
Two important results follow immediately from Theorem 3.3.
THEOREM 3.4.
PROOF.
AdJ (G,A)
Let B be basic in G.
Since 0
Adj (B,A)
/
B
/
G
/
G/B
/
0
Then, for arbitrary A, AdJ(G,A)
-" AdJ(B,A).
is pure exact, Theorem 3.3 yields
AdJ (G/B,A). But AdJ (G,B/A)
0, since Hom(G/B,A) is
torsion free.
COROLLARY 3 5
If B is basic in G, then
Hom(G,A)p (n) Hom(B,A)p (n)
for all
integers n >_ 0.
PROOF.
This follows from Theorem 3.4 and the observation that the Ulm invariants
75
HOMOMORPHISMS OF ABELIAN TORSION GROUPS
of an algebraically compact group are the same as the Ulm Invariants of its
adjusted part.
We are now ready to calculate the Ulm invarlants for Hom(G,A).
THE ULM INVARIANTS OF HOM<G,A).
4.
Let B
Let G and A be p-groups with B basic in G.
Bi=
E
Z(.pi..
S (i-l)
P
B where
i_Zl
i
Then
=-- i=l Horn (Bi ,A)
Hom(B,A)
--"
i_l
G
P
i-l)
Hm(Z(Pl)’A) i-i c i-i) A[pl]
P
It follows that
Hom(G,A) (n)
P
dim
dim
Hom(B,A) (n)
P
(pn(A[pi]))[P]/(pn+I(A[pi]))[P]
G i-l)
P
i=l
i 1 GPi-l) A[p i] P (n)
Observe that for m > n,
Z(p).
(pn(A[pm]))[p]
(pnA)[p]
n,
and for m
pn (A[pm])
0.
Thus
Alp i] (n)
P
0
ifi<n
r(pnA)
if i
n
+ 1
A(n)
if i > n
+ 1
p
r(pnA)
denotes the rank of
THEOREM 4.1.
dim
pnA.
where
Hence we have proved the following theorem.
If G and A are p-groups, then
Hom(G,A)p(n)
Z(p).
Z(p) + dim
Z
Z
G (n) r(pnA)
i--n+l G (i) A (n)
P
P
P
In Theorem 4. i, as in certain subsequent theorems, we prefer not to compute
the dimensions of the Z(p)-spaces involved; such computations are easy, but the
resulting expressions are clumsy.
5.
THE TORSION FREE NUMBER OF HOM(G,A).
The computation of
Hom(G,A)0(p)
will be done in three steps by the following
M.W. LEGG
76
LEMMA 5.1.
PROOF.
Let G and A be divisible.
Note that Hom(G,A)
-
Horn (Z
r(G)
Then
Hom(G,A)0(p)
r.
Z(p).
r (C) r (A)
dim
Z(p): Z Z(p))
r(A)
Z(p ))
Hom(Z(p), Z
Z(p )). Since dim Hom(Z(p), Z
H
r(A)
r(G)
r(A)
(cf. [2] or [3]), the result follows.
The final rank of a p-group G is defined to be fr(G)
r(A)
{r(pnG)]n
rain
0,I,2,...}
(cf. Fuchs [13).
Let G be reduced and A be divisible.
LEMMA 5.2.
Hom(G,A)o(p)
PROOF.
n
dim
Then
Z(p).
l
fr(G) r(A)
Hence we can
If fr(G) is finite, it is zero and the result is clear.
By Fuchs [I] we can write G
assume that fr(G) is infinite.
is bounded and
r(G 2)
fr(G 2)
we can assume that fr(G)
fr(G).
r(G).
Hence, since
G
G2, where G I
Hom(G2,A)0(p),
I
Hom(G,A)0(p)
Let B be basic in G such that r(G/B)
r(G).
Then 0 +B /G /G/B +0 pure exact and A divisible implies that
0 + Hom(G/B,A)
/
Hom(G,A)0(p)
Hom(G/B,A)0(p)
Hom(G,A)
a monomorphism from B to
Hom(G/B,A)
Since
Thus, since
yields
/
+
/
0 is split exact.
Hom(B,A)0(p).
Since
Thus
Hom(G/B,A)0 (p)
dim
r(G/B), there
r(B) <_r(G)
G/B, hence an epimorphism of Hom(G/B,A)
is torsion
Hom(G,A)0(p)
Hom(B,A)
onto
is
Hom(B,A).
Hom(G/B,A)o(p) >_Hom(B,A)0(p).
Hom(G,A)0 (p) Hom(G/B,A)0(p). This
free, it follows that
is infinite, we have
Z Z(p) on application of Lemma 5.1.
fr (G) r (A)
The final rank of a basic subgroup of a p-group G is an invariant of G and is
called the critical number of G.
LEMMA 5.3.
PROOF.
We shall denote it cr(G) (cf. Szele [63).
Let G and A be reduced.
Then
Hom(G,A)0(p)
Let B be basic in G and C basic in A.
Let cr(G)
dim
H
Z
Z(p).
cr(G) cr(A)
a and cr(A)
B.
HOMOMORPHISMS OF ABELIAN TORSION GROUPS
If
is finite it is zero and B is bounded.
It follows that
Hom(G,A)0(p
=
Hence we may take both
0.
IAI
Hom(G,A)0(p
and 8 to be infinite.
IcJ N
<
s
Hom(G,A)0(p)
"
<
IHom(G,A)
<
(Fuchs [i]).
a.
8
Thus G is bounded; hence so is Hom(G,A).
Similarily, if 8 is finite the result follows.
beginning of the previous proof, we can
Clearly
77
By remarks similar to those at the
assue
that r(B)
IHom(B,A)l.
Thus JHom(B,A)
<
IBI
,
’ )a
8a
Now
<(8
fr(B) and r(C)
ICI
fr(C).
8, and
so that
We will reverse this inequality by constructing 8
elements in
Hom(G,A) / (pHom(G,A) + tHom(G,A)).
Let B
E
B
i=l
Let
{bi’lJ3
E
C
and
{cljlj
IJil
that
IKil
e and
Ji}
e
for each n,
and C
i
i=n
can assume
that
{IKil}
j
J
i.3
e K
i=n
i}
be bases for B
IKil
IJi.3
of
j=l
and C
i
E
{Ji
IKij
i=l
s
i.
index B
i.
such that{
and K
i
index C
induces a homomorphism, say
Bi.3
,
’
{!}
3 j=l
Let
i.
j
:B
/
ij
C
i
to the generators of
l}j= 1
by
such
8j.
is non-decreasing,
Kij
let
Jij
i
of B.
j.
be a homomorphlsm induced by
Ci..3
Each sequence
from B to C, namely homomorphism that is
and mero on the other components B
and
IKij
{K}’i=l
of
and so that the common order of the generators indexed by
e
a map from the generators of
{j }=j=l
{Kij }i=l
denote
is greater than the order of the generators indexed by
Let J.
We note that,
respectively.
Choose a subsequence
j=
{Ji.
i
It will be clear from the proof that we
8.
> 1 for each i.
Next choose a subsequence
j=l
in the usual notation.
is non-decreaslng and
j=l
such that
i
j
{j}j=l
on
Call two such homomorphisms
equivalent if
3
’
3
for almost all
J.
B.ej
It follows that the cardinality of the set of equivalence classes is
J-I
and it is routine to show that
B ej
j=l 3
Bij
B
e
J
M.W. LEGG
78
is an epimorphism from G to B and
Now if
6 # 6’.
p
n
Suppose
6- 6’)
p
6’
6
n+l
py
+
6,6’ are two InequSvalentmaps, then
Hom(G,A) and
where y,T e
T
pnT
0.
Then
According to the construction, there is a generator, say c.i
of order greater than p
n
is a component of (6
6’)(x). Since
Clk
Cik
n+l
p
n. But pn(6
has
7(x)
6’)(x)
has
Thus there is an x e G such that
height zero,
pn( 6-
6’)(x)
height exactly
Hence no such y and
which gives a contradiction.
and 6’ are inequivalent maps from B to C, then
cosets in Hom(G,A)/(pHom(G,A)
Hm(G,A) O(p)-
tHom(G,A))
THEOREM 5.4.
Let G
+ tHom(G,A)).
T exist.
6
Thus
6’
and
8
We have shown that if
determine different
<_ IHom(G,A)/(pHom(G,A) +
The result follows.
If G and A are p-groups, then
Hom(G,A)0(p)
PROOF.
k
as a component of some element in the image of 6-6’.
G
dim
ldG
Hom(G,A)
-
Z(p) + dim
l
cr(G) cr(A)
and A =A
ldA.
Hom(G I,A I)
Z(p).
l
H
fr(G) r(dA)
Then
Hom(dG,dA).
Hom(G l,dA)
Fom the preceding three lemmas we obtain
Hom(G,A)o (p)
Z(p)
dim
+
dim
cr(G I) cr(A I)
Using the relations cr(G)
H
fr
cr(A)
cr(Gl)
Z(p) + dim
7.
(GI)
r (dA)
cr(Al)
fr(G)
7.
Z(p)
H
r(dG) r(dA)
fr(G I) + r(dG), the
result follows.
Since
Hom(G,A)
Hom(G,A)q(n)
is q-divisible for all primes q
are zero for such q.
# p,
Hom(G,A)0(p)
and each
Thus we can summarize our results in the following
theroem.
THEOREM 5.5.
If G and A are p-groups, then Hom(G,A) is algebraically compact.
Complete invariants for Hom(G,A) are
Hom(G,A)o(p)
dim
Z(p)
Z(p) + dim H
H
l
fr(G) r(dA)
cr(G) cr(A)
79
HOMOMORPHISMS OF ABELIAN TORSION GROUPS
and for n
0,I,2,...
Hom(G,A)p(n)
6.
dim
H
l
Gp (n) r(pnA)
Z(p) + dim
H
i--n+l
l
Z(p)
Gp(i) Ap(n)
CONCLUSION.
Finally, it is easy to see that Theorem 5.5 allows us to calculate the Ulm
invariants and the torsion free numbers for
Hom(X,Y), where X and Y are arbitrary
torsion groups
Let t X denote the p-primary component of X. Then Hom(X,Y) = H
q
P
Horn
where q ranges over the primes. Thus for each n Hom(X,Y) (n)
P
I
it follows that
Also, since
q
q
Hom(tqX,tqY),
(tpX, tpg )p (n).
tHom(tqX,tqY),
tHHom(tqX,tqY)
H Hom(tqX, tqg) / (pH Hom(tqX,tqY) + tg Hom(tqX,tqg))
q
q
q
Hom(tpX, tpY) / (pHom(tpX, tpY) + tHom(tpX,tpY))
We have the following theorem.
THEOREM 6.1.
compact.
If X and Y are torsion groups, then Hom(X,Y) is algebraically
Complete invariants for Hom(X,Y) are given by
Hom(tpX, tpY)0 (p)
(X,Y)p(n) Hom(tpX,TpY)p(n)
Hom(X,Y)0 (p)
and
Horn
where p ranges over the primes and n ranges over
the non-negatlve integers.
Hence the invarlants calculated in Theorem 5.5 characterize Hom(X,Y).
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Infinite Abelian
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