Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

§ 7.5 Finite Abelian Groups

Theorem 7.5.1. A finite abelian group can be expressed as a direct product of its Sylow p-subgroups.

Lemma 7.5.2. Let G be a finite abelian p-group, let a be an element of maximal order in G, and let b<a> be any coset of G/<a>. Then there exists d in G such that d<a> = b<a> and <a> <d> = {e}.

Lemma 7.5.3. Let G be a finite abelian p-group. If <a> is a cyclic subgroup of G of maximal order, then there exists a subgroup H with G <a> × H.

Theorem 7.5.4. [Fundamental Theorem of Finite Abelian Groups] Any finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Any two such decompositions have the same number of factors of each order.

Proposition 7.5.5. Let G be a finite abelian group. Then G is isomorphic to a direct product of cyclic groups

Zn1 × Zn2 × · · · × Znk

such that ni | ni-1 for i = 2,3, . . . ,k.

Corollary 7.5.6. Let G be a finite abelian group. If a is an element of maximal order in G, then the order of every element of G is a divisor of the order of a.

Theorem 7.5.8. Let p be an odd prime, let k be a positive integer, and let n = pk. Then Zn× is a cyclic group.

Theorem 7.5.10. If k 3, and n = 2k, then Zn× is isomorphic to the direct product of a cyclic group of order 2 and a cyclic group of order 2k-2.

Corollary 7.5.11. The group Zn× is cyclic if and only if n is of the form 2, 4, pk, or 2pk for an odd prime p.

In elementary number theory, an integer g is called a primitive root for the modulus n if Zn× is a cyclic group and [g]n is a generator for Zn×. Corollary 7.5.11 determines which moduli n have primitive roots. The proof of Theorem 7.5.8 shows how to find a generator for Zn×, where n = pk.

§ 7.5 Finite Abelian Groups: Solved problems

7. Find all abelian groups of order 108 (up to isomorphism).     Solution

8. Let G and H be finite abelian groups, and assume that G × G is isomorphic to H × H. Prove that G is isomorphic to H.     Solution

9. Let G be an abelian group which has 8 elements of order 3, 18 elements of order 9, and no other elements besides the identity. Find (with proof) the decomposition of G as a direct product of cyclic groups.     Solution

10. Let G be a finite abelian group such that |G| = 216. If | 6 G | = 6, determine G up to isomorphism.     Solution

11. Apply both structure theorems to give the two decompositions of the abelian group Z216×.     Solution

12. Let G and H be finite abelian groups, and assume that they have the following property. For each positive integer m, G and H have the same number of elements of order m. Prove that G and H are isomorphic.     Solution

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