# st1 - 2 Does there exist a nonabelian group of order 8 with...

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Math 4124 Monday, February 7 Sample First Test. Answer All Problems. Please Give Explanations For Your Answers. 1. Prove that there is no positive integer n such that S 3 × S 5 is isomorphic to S n . (8 points) 2. Let G = { A GL 2 ( R ) | AA t = I } , where A t denotes the transpose of A and I denotes the identity 2 by 2 matrix. Prove that G is a subgroup of GL 2 ( R ) . (You may assume that ( A t ) t = A and ( AB ) t = B t A t for all 2 by 2 matrices A , B .) (8 points) 3. Let G be a group. Prove that the formula ( g , h ) · x = gxh - 1 for g , h , x G , deﬁnes an action of G × G on G . Show further that if Z ( G ) 6 = 1, then there exists a nonidentity element which acts trivially on G (i.e. there exists 1 6 = k G × G such that k · x = x for all x G ). (9 points) 4. Let G be a cyclic group of order 8. Prove that G has exactly one element of order
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Unformatted text preview: 2. Does there exist a nonabelian group of order 8 with this property? (Justify your answer!) (8 points) 5. Let G be a group and let a ∈ G . Prove that C G ( h a i ) = C G ( { a } ) . (9 points) 6. Let G be a ﬁnite cyclic group and let N ± G . Prove that there exists H ≤ G such that H ∼ = G / N . Does this result remain true without the hypothesis that G is ﬁnite? (8 points) Test on Monday, February 21. Material as far as section 3.3 approximately . One problem will be identical to one of the ungraded homework problems. Review session on Sunday, February 20 at 4:45 p.m. in McBryde 210....
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