Page 125 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes The function defined in the example is a special case of a more general result that is usually
referred to as the Chinese remainder theorem (this result is given more generally for rings. The
proof of the next proposition makes use of the same function.
Proposition: If k = mn, where m and n are relatively prime integers, then Z is isomorphic to
k
Z  Z .
n
m
Outline of the Proof: Define f : Z  Z  Z by f([x] ) = ([x] , [x] ), for all x  Z. Here I have been
m
k
k
n
m
n
a bit more careful, by using [x] to denote the congruence class of x, modulo k. It is not hard to
k
show that f preserves addition. The sets Z and Z  Z are finite and have the same number of
n
k
m
elements, so f is one-to-one iff it is onto, and therefore proving one of these conditions will give
the other. (Actually, it isn’t hard to see how to prove both conditions.) Showing that f is one-to-
one depends on the fact that if x is an integer having both m and n as factors, then it must have
mn as a factor since m and n are relatively prime. On the other hand, the usual statement of the
Chinese remainder theorem is precisely the condition that f is an onto function.
Corollary: Any finite cyclic group is isomorphic to a direct sum of cyclic groups of prime power
order.
The corollary depends on an important result in Z: every positive integer can be factored into a
product of prime numbers. Grouping the primes together, the proof of the corollary uses induction
on the number of distinct primes in the factorization.
This basic result has implications for all finite groups. The cyclic group Z also has a ring
n
structure, and the isomorphism that proves the corollary is actually an isomorphism of rings,
not just of finite abelian groups. To use this observation, suppose that A is a finite finite abelian
group. Let n be the smallest positive integer such that na = 0 for all a  A. (This number might be
familiar to you in reference to a multiplicative group G, where it is called the exponent of the
group, and is the smallest positive integer n such that g = 1 for all g  G.)
n
You can check that because na = 0 for all a  A, we can actually give A the structure of a
Z -module.
n
Next we can apply a general result that if a ring R can be written as a direct sum R = I  . . .I n
1
of two-sided ideals, then each I is a ring in its own right, and every left R-module M splits up
j
into a direct sum M . . . M , where M is a module over I . Applying this to Z , we can write
1
n
j
n
j
Z as a direct sum of rings of the form Z , where p is a prime, and then the group A breaks up
k
p
n
into A  . . .  A , where each A is a p-group, for some prime p. (Recall that a group G is a
1
n
j
p-group if every element of G has order p.) This argument proves the next lemma. (You can also
prove it using Sylow subgroups, if you know about them.)
Every finite abelian group can be written as a direct sum of p-groups.
The decomposition into p-groups occurs in one and only one way. Then it is possible to prove
that each of the p-groups splits up into cyclic groups of prime power order, and so we have the
following fundamental structure theorem for finite abelian groups.
Theorem 1: Any finite abelian group is isomorphic to a direct sum of cyclic groups of prime
power order.
A proof of the fundamental structure theorem, let us first discuss some of the directions it
suggests for module theory. First of all, the hope was to construct finite abelian groups out of
ones of prime order, not prime power order. The only way to do this is to stack them on top of
each other, instead of having a direct sum in which the simple groups are lined up one beside the
other. To see what I mean by “stacking” the groups, think of Z and its subgroups Z  2Z (0).
4
4
4
It might be better to picture them vertically.

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