113 HW 4_Math 113

# 113 HW 4_Math 113 - Mathematics 113 Homework 4 Curtis...

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Mathematics 113: Homework # 4 Curtis Tekell Dr. Poirier October 4, 2010 Exercise 1. Let G be a group acting on a set X and let K be the kernel of this action and G x be the stabalizer of x X of the action. Then K,G x G . Proof. Suppose a,b K . Then b - 1 x = ( b - 2 b ) x = b - 2 ( bx ) = b - 2 (1) = 1, and thus ab - 1 x = a ( b - 1 ) = a 1 = 1, and ab - 1 K , so K G . Now suppose a,b G x . Then b - 1 x = b - 1 ( bx ) = ( b - 1 b ) x = x , and thus ab - 1 x = ax = x , and ab - 1 G x , so G x G . Exercise 2. Let H G . (a) H N ( H ) , this is not true if H ± G . (b) H C ( H ) if and only if H is abelian. Proof. (a) Let h H . Then clearly for any k H , khk - 1 H , as H G , so k N ( H ), and thus H N ( H ). We now exhibit a example to show that this does not hold if H ± G . Let G = GL 2 ( R ), and let H = ± A = ² 1 1 0 1 ³ ,B = ² 1 0 1 1 ³´ . Then one can check that ABA - 1 = ² 2 0 1 0 ³ / H , thus A / N ( H ), and H ± N ( H ). (b) If H is abelian, clearly H C ( H ). Conversely, if H is abelian, all of its elements commute with each other, so H C ( H ). 1

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Exercise 3. (a) Show that the group of rotations G of a cube is isomorphic to S 4 . (b) Show that the group of rotations G of an octahedron is also isomorphic to S 4 . Proof.
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## This note was uploaded on 10/15/2011 for the course CHEM 1 taught by Professor Gelfand during the Summer '11 term at Solano Community College.

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113 HW 4_Math 113 - Mathematics 113 Homework 4 Curtis...

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