Page 126 - DMTH403_ABSTRACT_ALGEBRA

Unit 10: Finite Abelian Groups

Z Notes
4
|

2Z 4
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(0)
The subgroup 2Z = {0, 2}  Z2 is simple, and so is the factor module Z /2Z  Z . This having Z 2
4
2
4
4
stacked on top of Z , and the group is structured so tightly that you can’t even find an isomorphism
2
to rearrange the factors.
A module M is said to have a composition series of length n if there is a chain of submodules M
= M  M  . . .  M = (0) for which each factor module M /M is a simple module. Thus, we
n
i
i–1
0
1
would say that Z has a composition series of length 2. This gives a measurement that equals the
4
dimension, in the case of a vector space. It is also true that the length of a cyclic group of order
p is precisely n. It can be shown that if M has a composition series of length n, then every other
n
composition series also has length n, so this is an invariant of the module. Furthermore, the
same simple modules show up in both series, with the same multiplicity.
The idea of a composition series is related to two other conditions on modules. A module is said
to satisfy the ascending chain condition, or ACC, if it has no infinite chain of ascending submodules;
it is said to satisfy the descending chain condition, or DCC, if it has no infinite chain of descending
submodules. Modules satisfying these conditions are called Noetherian or Artinian, respectively.
A module has finite length iff it satisfies both the ACC and DCC. As an example to keep in mind,
let’s look at the ring of integers, which has ACC but not DCC. Since mZ  nZ iff n | m,
generators get smaller as you go up in Z, and larger as you go down. Any set of positive integers
has a smallest element, so we can’t have any infinite ascending chains, but, for example, we can
construct the infinite descending chain 2Z  4Z 8Z ... .
The cyclic groups of prime power order play a crucial role in the structure of finite abelian
groups precisely because they cannot be split up any further. A module M can be expressed as a
direct sum of two submodules M and M iff M  M = (0) and M + M = M. In the case of a cyclic
2
1
2
1
1
2
group of prime power order, the subgroups form a descending chain, and so any two nonzero
subgroups have a nonzero intersection. A module is called indecomposable if it cannot be
written as a direct sum of two nonzero submodules. With this terminology, the cyclic groups of
prime power order are precisely the indecomposable finite abelian groups. The major results in
this direction are (the Krull-Schmidt theorem), which show that any module with finite length
can be written as a direct sum of indecomposable submodules, and this decomposition is unique
up to isomorphism and the order of the summands.
After this rather lengthy preview, or review, as the case may be, it is time to move on to study
general rings and modules. The next results present a proof of the structure theorem for finite
abelian groups, but you should feel free to skip them.
Lemma: Let A be a finite abelian p-group.
(a) Let a  A be an element of maximal order, and let b + Za be any coset of A/Za. Then there
exists d  A such that d + Za = b + Za and Zd  Za = (0).
(b) Let a  A be an element of maximal order. Then there exists a subgroup B with A  Za  B.
Proof: (a) The outline of part (a) is to let s be the smallest positive integer such that sb  Za. Then
we solve the equation sb = sx for elements x  Za and let d = b – x.

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