Unformatted text preview: order p 7 for any prime p . Corollary 3.6.11. (Cauchy’s theorem for Finite Abelian Groups) If G is a ﬁnite abelian group and p is a prime dividing the order of G , then G has an element of order p . Proof. Since G is a direct product of cyclic groups, p divides the order of some cyclic subgroup C of G , and C has an element of order p by Proposition 2.2.32 . n Let p be a prime. Recall that a group G is called a p –group if every element has ﬁnite order and the order of every element is a power of p . Corollary 3.6.12. A ﬁnite abelian group is a p –group if, and only if, its order is a power of p . Proof. If a ﬁnite group G has order p k , then every element has order a power of p , by Lagrange’s theorem. Conversely, a ﬁnite abelian p –group G has no element of order q , for any prime q diffferent from p . According...
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- Fall '08
- Algebra, Abelian group, Cyclic group, Finite Abelian Groups, abelian groups