Abstract Algebra Theory and Applications - Computer Science ...

11.1 FINITE ABELIAN GROUPS 193it follows that g a ib imust be in G i . Let g i = g a ib i. Then g = g 1 · · · g n andG i ∩ G j = {e} for i ≠ j.To show uniqueness, suppose thatwith h i ∈ G i . Theng = g 1 · · · g n = h 1 · · · h ne = (g 1 · · · g n )(h 1 · · · h n ) −1 = g 1 h −11 · · · g n h −1n .The order of g i h −1iis a power of p i ; hence, the order of g 1 h −11 · · · g n h −1n is theleast common multiple of the orders of the g i h −1i. This must be 1, since theorder of the identity is 1. Therefore, |g i h −1i| = 1 or g i = h i for i = 1, . . . , n.□We shall now state the Fundamental Theorem of Finite Abelian Groups.Theorem 11.3 (Fundamental Theorem of Finite Abelian Groups)Every finite abelian group G is isomorphic to a direct product of cyclic groupsof the formZ pα 11× Z pα 22× · · · × Z pαnnwhere the p i ’s are primes (not necessarily distinct).Example 3. Suppose that we wish to classify all abelian groups of order540 = 2 2 · 3 3 · 5. The Fundamental Theorem of Finite Abelian Groups tellsus that we have the following six possibilities.• Z 2 × Z 2 × Z 3 × Z 3 × Z 3 × Z 5 ;• Z 2 × Z 2 × Z 3 × Z 9 × Z 5 ;• Z 2 × Z 2 × Z 27 × Z 5 ;• Z 4 × Z 3 × Z 3 × Z 3 × Z 5 ;• Z 4 × Z 3 × Z 9 × Z 5 ;• Z 4 × Z 27 × Z 5 .The proof of the Fundamental Theorem relies on the following lemma.Lemma 11.4 Let G be a finite abelian p-group and suppose that g ∈ G hasmaximal order. Then G can be written as 〈g〉×H for some subgroup H of G.