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# st2 - Math 4124 Monday March 14 Sample Second Test Answer...

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Math 4124 Monday, March 14 Sample Second Test. Answer All Problems. Please Give Explanations For Your Answers. 1. Let G be a finite group, let H G , and let N G . If ( | N | , | G / H | ) = 1, prove that N H . (10 points) 2. Let G = S 4 and let π : G S G denote the regular representation of G (so S G = S 24 ). If t G is a transposition, what is the cycle decomposition of π t ? Use this to show that G is isomorphic to a subgroup of A 24 . (10 points) 3. Compute the orders of the conjugacy class and the centralizer of (1 2 3)(4 5 6)(7 8) in S 10 . (10 points) 4. Let p be a prime and let G be a nonabelian group of order p 4 . Prove that | Z ( G ) | = p or p 2 . (10 points) 5. Show that
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