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Section13.1Finite Abelian Groups

In our investigation of cyclic groups we found that every group of prime order was isomorphic to \({\mathbb Z}_p\), where \(p\) was a prime number. We also determined that \({\mathbb Z}_{mn} \cong {\mathbb Z}_m \times {\mathbb Z}_n\) when \(\gcd(m, n) =1\). In fact, much more is true. Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order; that is, every finite abelian group is isomorphic to a group of the type \begin{equation*}{\mathbb Z}_{p_1^{\alpha_1}} \times \cdots \times {\mathbb Z}_{p_n^{\alpha_n}},\end{equation*} where each \(p_k\) is prime (not necessarily distinct).

First, let us examine a slight generalization of finite abelian groups. Suppose that \(G\) is a group and let \(\{ g_i\}\) be a set of elements in \(G\), where \(i\) is in some index set \(I\) (not necessarily finite). The smallest subgroup of \(G\) containing all of the \(g_i\)'s is the subgroup of \(G\) generated by the \(g_i\)'s. If this subgroup of \(G\) is in fact all of \(G\), then \(G\) is generated by the set \(\{g_i : i \in I \}\). In this case the \(g_i\)'s are said to be the generators of \(G\). If there is a finite set \(\{ g_i : i \in I \}\) that generates \(G\), then \(G\) is finitely generated.

Example13.1

Obviously, all finite groups are finitely generated. For example, the group \(S_3\) is generated by the permutations \((12)\) and \((123)\). The group \({\mathbb Z} \times {\mathbb Z}_n\) is an infinite group but is finitely generated by \(\{ (1,0), (0,1) \}\).

Example13.2

Not all groups are finitely generated. Consider the rational numbers \({\mathbb Q}\) under the operation of addition. Suppose that \({\mathbb Q}\) is finitely generated with generators \(p_1/q_1, \ldots, p_n/q_n\), where each \(p_i/q_i\) is a fraction expressed in its lowest terms. Let \(p\) be some prime that does not divide any of the denominators \(q_1, \ldots, q_n\). We claim that \(1/p\) cannot be in the subgroup of \({\mathbb Q}\) that is generated by \(p_1/q_1, \ldots, p_n/q_n\), since \(p\) does not divide the denominator of any element in this subgroup. This fact is easy to see since the sum of any two generators is \begin{equation*}p_i / q_i + p_j / q_j = (p_i q_j + p_j q_i)/(q_i q_j).\end{equation*}

The reason that powers of a fixed \(g_i\) may occur several times in the product is that we may have a nonabelian group. However, if the group is abelian, then the \(g_i\)s need occur only once. For example, a product such as \(a^{-3} b^5 a^7\) in an abelian group could always be simplified (in this case, to \(a^4 b^5\)).

Now let us restrict our attention to finite abelian groups. We can express any finite abelian group as a finite direct product of cyclic groups. More specifically, letting \(p\) be prime, we define a group \(G\) to be a \(p\)-group if every element in \(G\) has as its order a power of \(p\). For example, both \({\mathbb Z}_2 \times {\mathbb Z}_2\) and \({\mathbb Z}_4\) are \(2\)-groups, whereas \({\mathbb Z}_{27}\) is a \(3\)-group. We shall prove the Fundamental Theorem of Finite Abelian Groups which tells us that every finite abelian group is isomorphic to a direct product of cyclic \(p\)-groups.

Example13.5

Suppose that we wish to classify all abelian groups of order \(540=2^2 \cdot 3^3 \cdot 5\). The Fundamental Theorem of Finite Abelian Groups tells us that we have the following six possibilities.

  • \({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_3 \times {\mathbb Z}_3 \times {\mathbb Z}_3 \times {\mathbb Z}_5\);

  • \({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_3 \times {\mathbb Z}_9 \times {\mathbb Z}_5\);

  • \({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_{27} \times {\mathbb Z}_5\);

  • \({\mathbb Z}_4 \times {\mathbb Z}_3 \times {\mathbb Z}_3 \times {\mathbb Z}_3 \times {\mathbb Z}_5\);

  • \({\mathbb Z}_4 \times {\mathbb Z}_3 \times {\mathbb Z}_9 \times {\mathbb Z}_5\);

  • \({\mathbb Z}_4 \times {\mathbb Z}_{27} \times {\mathbb Z}_5\).

The proof of the Fundamental Theorem of Finite Abelian Groups depends on several lemmas.

Lemma 13.6 is a special case of Cauchy's Theorem (Theorem 15.1, which states that if \(G\) be a finite group and \(p\) a prime such that \(p\) divides the order of \(G\), then \(G\) contains a subgroup of order \(p\). We will prove Cauchy's Theorem in Chapter 15.

If remains for us to determine the possible structure of each \(p_i\)-group \(G_i\) in Lemma 13.8.

The proof of the Fundamental Theorem of Finite Abelian Groups follows very quickly from Lemma 13.9. Suppose that \(G\) is a finite abelian group and let \(g\) be an element of maximal order in \(G\). If \(\langle g \rangle = G\), then we are done; otherwise, \(G \cong {\mathbb Z}_{|g|} \times H\) for some subgroup \(H\) contained in \(G\) by the lemma. Since \(|H| \lt |G|\), we can apply mathematical induction.

We now state the more general theorem for all finitely generated abelian groups. The proof of this theorem can be found in any of the references at the end of this chapter.