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The Fundamental Theorem of Abelian GroupsEarlier we mentioned a broad goal of the theory of finite groups is to arrive at a classification of all finite groups. In this section we will achieve a much more modest goal: We will classify all finite abelian groups. Our main result will be the fundamental theorem of abelian groups (FT), which asserts that a finite abelian group is isomorphic to a direct product of cyclic groups. We will be able to pin down the structure of a finite abelian group further by specifying to some extent the cyclic groups which occur. Theorem 1 (FT): Let G be a finite abelian group. Then G is isomorphic to a direct product of cyclic subgroups. Proof: Suppose that G has order n. The theorem is clear if
G/C0
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Each Hi is of the form Ci/C0, where Ci is a subgroup of G which contains C0. Since Hi is cyclic,
G
![]() ![]()
for some choice of xi such that
gC0 = (x1C0)a1...(xsC0)as = x1a1...xsasC0.
Thus,
g = c0x1a1...xsas
for some
g = x0a0(x1a1...xsas),
and g is the product of an element of [x0] and an element of [x1...xs]. In order to apply Theorem 3 of the section on direct products, we must prove that (3)
[x0]
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This is the heart of the proof. Let ai denote the order of xi. Let us first show the order of xiC0 (in G/C0) is also ai. Let bi denote the order of xiC0. Since
bi|ai
Since (xiC0)bi = C0, we see that
xibi = x0c, 0 < c < bi.
The order of xibi is
ai = bir/(r,c).
On the other hand, since x0 has maximal order in G,
bi/(r,c) < r which is equivalent to bi < (r,c).
If
bi < c,
which contradicts (5). Therefore,
(x1C0)
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By what we have proved above,
[x1,...,xs]
![]() ![]() ![]() so that by (2),
G
![]() ![]() ![]() and G is a direct product of cyclic groups. Corollary 2: Let G be a finite abelian group. Then there exist positive integers n1,...,nt such that Proof: Every cyclic group is isomorphic to Zn for some positive integer n.
Corollary 3: Let G be a finite abelian group. Then there exists a set of prime powers Proof: Let n1,...nt be as in Corollary 2, and let
ni =
![]() ![]() Then by Corollary 5 of the section on direct products, Note that
q1 < q2 < ... < qv
Let (8)
qiai1,qiai2,...,qiaij(i)
be the powers of the prime qi (not necessarily distinct) appearing in the decomposition of Corollary 3, arranged so that
0 < ai1 < ai2 < ... < aij(i).
Define Then Corollary 3 and the above remark imply that (9)
G
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It is clear that every element of G(qi) has order a power of qi. Moreover, from (9), we see that the set of all element of G(q1 Corollary 4: Let H(qi) denote the subgroup of G consisting of all elements whose order is a power of qi. Then and
G
![]() ![]() ![]() If q is a prime, then H(q) is called the q-primary part of G. Let us consider an example. Let Therefore,
The 2-primary part of G is Definition 5: The prime powers
qiaij [1 < j < j(i), 1 < i < v]
are called the elementary divisors of G. It is clear that if we are given the elementary divisors of G, then it is possible to determine G up to isomorphism using (9). Theorem 6: The elementary divisors of G are uniquely determined by G. In other words, the decomposition (9) of G is unique. Proof: The primes q1,...,qv are uniquely determined as the distinct primes dividing the order of G. Let ![]()
It suffices to show that ![]()
so that order of Hqa-1 is q ![]() Thus, the order of Hqa-2 is
q
![]() ![]() Thus, Example 1: Let us determine up to isomorphism all abelian groups of order 16. Since (a) 24. (b) 2, 23. (c) 2, 2, 22. (d) 2, 2, 2, 2. (e) 22, 22. The corresponding abelian groups are (a) Z16. None of the groups (a)-(e) are isomorphic to one another by Theorem 6. This example suggests the following generalization. Let p be a prime, m a positive integer. Let us determine the abelian groups of order pm. We know that every such groups is uniquely specified by its elementary divisors
pa1,pa2,...,pas,
where
a1 < a2 < ... < as, a1 + a2 + ... + as = m.
The last property comes from the fact that (10)has order pa1·pa2...pas. Thus, to each abelian group of order pm is associated the set {a1,...,as} of positive integers such that a1 + ... + as = m, a1 < a2 < ... < as. Such a set is called a partition of m. Conversely, to each partition {a1,...,as} of m there corresponds an abelian group of order pm - the group (10). Moreover, by Theorem 6, distinct partitions of m correspond to nonisomorphic groups. Let p(m) denote the number of distinct partitions of m. Then, we see that the number of nonisomorphic abelian groups of order pm is p(m). In the above example, p = 2, m = 4. The partitions of 4 are
{4}, {1,3}, {1,1,2}, {1,1,1,1}, {2,2}.
Thus, there are five nonisomorphic abelian groups of order 24, as we discovered. We proved earlier that the number of nonisomorphic groups of order n is at most nn2. If The problem of determining the law of growth of the partition function p(m) has been solved completely, but only in the twentieth century, and belongs to a chapter of contemporary mathematics called additive number theory. Although many results about partitions had been discovered as early as the eighteenth century by Euler and Lagrange, it was not until the invention of the circle method by Hardy and Ramanujan in 1918 that any real progress was made in determining the order of magnitude of p(m). Hardy and Ramanujan proved that (11)![]() In other words, for large m, p(m) is approximately equal to ![]() We leave it to the reader to demonstrate the this quantity is much smaller that pmp2m, in the sense that ![]() The proof of (11) is very difficult and relies on analysis. Much more precise results than (11) are known. In fact in 1936 Rademacher found a beautiful exact formula for p(m) in terms of an infinite series. Let us close this section with a very simple, but useful application of the fundamental theorem. Proposition 7: Let G be an abelian group whose order is divisible by a prime p. Then G contains an element of order p. Proof: Without loss of generality assume G to be of the form Then
(pa1-1,pa2-2,...,pat-1,0,0,...,0)
has order p. |
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