ProblemSet7

# ProblemSet7 - subgroup of S n(b Prove that an inﬁnite...

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Homework 7 Due: Wednesday 3/24/10 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Prove that a ﬁnite p -group G is simple if and only if | G | = p . 2. (a) If H is a subgroup of G and if x H prove that C H ( x ) = H C G ( x ). (b) If H is a subgroup of index 2 in a ﬁnite group G and if x H , prove that either | x H | = | x G | or | x H | = 1 2 | x G | , where x H is the conjugacy class of x in H . 3. (a) Prove that if a simple group G has a subgroup of index n , then G is isomorphic to a
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Unformatted text preview: subgroup of S n . (b) Prove that an inﬁnite simple group has no subgroups of ﬁnite index n > 1. 4. Leg G be a group with | G | = mp , where p is prime and 1 < m < p . Prove that G is not simple. 5. If n ≥ 3, prove that A n is the only subgroup of S n of order 1 2 n !. 6. Show that S 4 has a subgroup isomorphic to D 4 . 7. Prove that A 5 is a group of order 60 having no subgroup of order 30....
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