Homework2 - R(b Prove that the center of G is trivial(c Let...

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Homework 2 Math 332, Spring 2010 These problems must be written up in L A T E X, and are due next Thursday, February 11. 1. Prove that every finite group with an even number of elements has at least one element of order two. 2. Let G be a group, and let a be a fixed element of G . Define a new binary operation * on G by x * y = xay for all x,y G . Prove that G forms a group under the operation * . 3. Let G = ±² a b 0 1 ³ : a,b R and a 6 = 0 ´ . (a) Use the two-step subgroup test to prove that G is a subgroup of GL (2 ,
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Unformatted text preview: R ). (b) Prove that the center of G is trivial. (c) Let x = ² a b 0 1 ³ and y = ² c d 0 1 ³ be two elements of G . Assuming that neither is the identity element, prove that x and y are conjugate in G if and only if a = c . 4. Let G be a group, and let a be an element of G . The centralizer of a in G is the set C ( a ) = { g ∈ G | ga = ag } . Use the two-step subgroup test to prove that C ( a ) is a subgroup of G . 1...
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