Back  Table of Contents 

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/C_{0} H_{1} H_{2} ... H_{s}.
Each H_{i} is of the form C_{i}/C_{0}, where C_{i} is a subgroup of G which contains C_{0}. Since H_{i} is cyclic,
G [x_{0}] [x_{1},...x_{s}].
gC_{0} = (x_{1}C_{0})^{a1}...(x_{s}C_{0})^{as} = x_{1}^{a1}...x_{s}^{as}C_{0}.
Thus,
g = c_{0}x_{1}^{a1}...x_{s}^{as}
for some
g = x_{0}^{a0}(x_{1}^{a1}...x_{s}^{as}),
and g is the product of an element of [x_{0}] and an element of [x_{1}...x_{s}]. In order to apply Theorem 3 of the section on direct products, we must prove that (3)
[x_{0}] [x_{1},...,x_{s}] = {1}.
This is the heart of the proof. Let a_{i} denote the order of x_{i}. Let us first show the order of x_{i}C_{0} (in G/C_{0}) is also a_{i}. Let b_{i} denote the order of x_{i}C_{0}. Since
b_{i}a_{i}
Since (x_{i}C_{0})^{bi} = C_{0}, we see that
x_{i}^{bi} = x_{0}^{c}, 0 < c < b_{i}.
The order of x_{i}^{bi} is
a_{i} = b_{i}r/(r,c).
On the other hand, since x_{0} has maximal order in G,
b_{i}/(r,c) < r which is equivalent to b_{i} < (r,c).
If
b_{i} < c,
which contradicts (5). Therefore,
(x_{1}C_{0})^{1}...(x_{s}C_{0})^{s} = x_{1}^{1}...x_{s}^{s}C_{0}, 0 < _{i} < a_{i}.
By what we have proved above,
[x_{1},...,x_{s}] [y_{1}] ... [y_{t}],
so that by (2),
G [x_{1}] [y_{1}] [y_{t}],
and G is a direct product of cyclic groups. Corollary 2: Let G be a finite abelian group. Then there exist positive integers n_{1},...,n_{t} such that Proof: Every cyclic group is isomorphic to Z_{n} for some positive integer n. Corollary 3: Let G be a finite abelian group. Then there exists a set of prime powers ,,..., (not necessarily distinct) such that Proof: Let n_{1},...n_{t} be as in Corollary 2, and let
n_{i} = ... (1 < i < t).
Then by Corollary 5 of the section on direct products, Note that
q_{1} < q_{2} < ... < q_{v}
Let (8)
q_{i}^{ai1},q_{i}^{ai2},...,q_{i}^{aij(i)}
be the powers of the prime q_{i} (not necessarily distinct) appearing in the decomposition of Corollary 3, arranged so that
0 < a_{i1} < a_{i2} < ... < a_{ij(i)}.
Define
G(q_{i}) = Z_{} Z ....
Then Corollary 3 and the above remark imply that (9)
G G(q_{1}) G(q_{2}) ... G(q_{v}).
It is clear that every element of G(q_{i}) has order a power of q_{i}. Moreover, from (9), we see that the set of all element of G(q_{1} ... G(q_{v}) whose order is a power of q_{i} is {1} {1} ... G(q_{i}) ... {1} G(q_{i}). Thus, we have Corollary 4: Let H(q_{i}) denote the subgroup of G consisting of all elements whose order is a power of q_{i}. Then and
G H(q_{i}) ... H(q_{v}).
If q is a prime, then H(q) is called the qprimary part of G. Let us consider an example. Let Therefore, The 2primary part of G is Z_{2}, while the 5primary part of G is Z_{5} Z_{52}. Definition 5: The prime powers
q_{i}^{aij} [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 q_{1},...,q_{v} are uniquely determined as the distinct primes dividing the order of G. Let
It suffices to show that (1), (2),...,(a) depend only on G and q. If b is a positive integer, then set so that order of H^{qa1} is q^{(a)}. Thus, (a) is determined only by G, q, and a. Next note that Thus, the order of H^{qa2} is
q^{(a1)+2(a)}.
Thus, (a  1) depends only on G, q, and a  1. Proceeding in this way, we see that (1),...,(a) are determined uniquely by G and q. Example 1: Let us determine up to isomorphism all abelian groups of order 16. Since (a) 2^{4}. (b) 2, 2^{3}. (c) 2, 2, 2^{2}. (d) 2, 2, 2, 2. (e) 2^{2}, 2^{2}. The corresponding abelian groups are (a) Z_{16}. 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 p^{m}. We know that every such groups is uniquely specified by its elementary divisors
p^{a1},p^{a2},...,p^{as},
where
a_{1} < a_{2} < ... < a_{s}, a_{1} + a_{2} + ... + a_{s} = m.
The last property comes from the fact that (10)has order p^{a1}·p^{a2}...p^{as}. Thus, to each abelian group of order p^{m} is associated the set {a_{1},...,a_{s}} of positive integers such that a_{1} + ... + a_{s} = m, a_{1} < a_{2} < ... < a_{s}. Such a set is called a partition of m. Conversely, to each partition {a_{1},...,a_{s}} of m there corresponds an abelian group of order p^{m}  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 p^{m} 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 2^{4}, as we discovered. We proved earlier that the number of nonisomorphic groups of order n is at most n^{n2}. 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 p^{mp2m}, 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
(p^{a11},p^{a22},...,p^{at1},0,0,...,0)
has order p. 
Back  Table of Contents 