College Algebra Exam Review 190

# College Algebra Exam Review 190 - order p 7 for any prime p...

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200 3. PRODUCTS OF GROUPS Example 3.6.8. Every abelian groups of order 32 is isomorphic to one of the following: Z 32 , Z 16 Z 2 , Z 8 Z 4 , Z 8 Z 2 Z 2 , Z 4 Z 4 Z 2 , Z 4 Z 2 Z 2 Z 2 , Z 2 Z 2 Z 2 Z 2 Z 2 . Definition 3.6.9. (a) A partition of a natural number n is a sequence of natural num- bers n 1 n 2 n s such that P i n i D n . (b) Let G be an abelian group of order p n . There exist uniquely determined partition .n 1 ; n 2 ; : : : ; n k / of n such that G Š Z p n 1 Z p ns . The partition is called the type of G . The type of an abelian group of prime power order determines the group up to isomorphism. The number of different isomorphism classes of abelian groups of order p n is the number of partitions of n . The number does not depend on p . Example 3.6.10. For example, the distinct partitions of 7 are (7), (6, 1), (5, 2), (5, 1, 1), (4, 3), (4, 2, 1), (4, 1, 1, 1), (3, 3, 1), (3, 2, 2), (3, 2, 1, 1), (3, 1, 1, 1, 1), (2, 2, 2, 1), (2, 2, 1, 1, 1), (2, 1, 1, 1, 1, 1), and (1, 1, 1, 1, 1, 1, 1). So there are 15 different isomorphism classes of abelian groups of order
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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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