Math 150a: Modern Algebra Homework 10 This problem set is due Friday, December 7. Do problems 5.6.3, 5.7.3 (using the counting formula and the stabilizer of a face), 5.7.5, 6.1.6, 6.2.4, 6.2.7, 6.3.2 (left multiplication only), 6.4.2, 6.4.5, and 6.4.12, in addition to the following: GK1. Let G be the subgroup of S 7 generated by a = ( 1 2 3 4 5 6 7 ) and b = ( 2 3 5 )( 4 7 6 ) . As part (a) will show, G is a dihedral-ish group in which the reflection f is replaced by an element of order 3. a. Show that bab − 1 = a 2 . (This is using my multiplication convention in S n ; otherwise you get bab − 1 = a 4 .) Using this, show that A = a a A is a normal subgroup of G and that it is complemented by B = a b A . How many elements are in G ? Is B normal? b. Show that A is normal and Sylow and show that B is Sylow. How many conjugates does B have? (To do this problem it helps enormously to make a standard form for elements of G using the fact that G = AB .) GK2. Let p be prime and let C p × C
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