# mar28 - Math 4124 Monday, March 28 March 28, Ungraded...

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Math 4124 Monday, March 28 March 28, Ungraded Homework Prove that a group of order 765 is abelian. Let G be a group of order 765 = 9 * 5 * 17. We need to prove that G is abelian. It is usually best to consider the largest prime dividing the order of the group ﬁrst, so we will ﬁrst consider the number of Sylow 17-subgroups. This is congruent to 1 mod 17 and divides 45, so the only possibility is 1. Thus G has a normal 17-subgroup, which we shall call A . Now we cannot immediately assert that the number of Sylow 5-subgroups is 1, because 51 appears possible. Instead we consider G / A , a group of order 765 / 17 = 45. Since the number of Sylow 5-subgroups is congruent to 1 mod 5 and divides 9, we see that G / A has a normal Sylow 5-subgroup P / A , where P ± G and | P | = 5 * 17 = 85. The number of Sylow 5-subgroups of P is congruent to 1 mod 5 and divides 17, so must be 1 and we see that P has a normal Sylow 5-subgroup B . Now B ± P ± G does not imply B ± G in general, but it does here because

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## This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

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mar28 - Math 4124 Monday, March 28 March 28, Ungraded...

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