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# rev2 - Math 4124 Wednesday March 30 Second Test Review The...

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Math 4124 Wednesday, March 30 Second Test Review The test will cover sections 3.3, 3.5 and chapter 4. Topics will include 1. If G is a group, H G and N ± G , then HN / N = H / H N . If furthermore K ± G and K N , then G / K N / K = G / N . 2. Transpositions and the alternating group. A n is a normal subgroup of index 2 in S n ( n 2). 3. Groups acting on sets. Orbits and stabilizers. 4. The size of the conjugacy class containing x G is | G / C G ( x ) | . 5. Left regular representation, conjugation action, conjugacy classes. 6. If | G / H | = n , then there is a homomorphism θ : G S n such that ker θ H . 7. If p is the smallest prime dividing | G | and | G / H | = p , then H ± G . 8. If | G | = p n where p is a prime and n 1, then Z ( G ) 6 = 1. 9. If | G | = p 2 where p is a prime, then G = Z / p 2 Z or G = Z / p Z × Z / p Z . 10. Aut ( G ) , an automorphism is determined by its affect on a generating set. 11.

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rev2 - Math 4124 Wednesday March 30 Second Test Review The...

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