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Unformatted text preview: H,N ≤ G and N C G , then (a) HN is a subgroup of G ; (b) H ∩ N is a normal subgroup of H ; (c) H/ ( H ∩ N ) ∼ = HN/N . 4. Let G be a group that contains an element x ∈ G that has exactly two conjugates. Prove that G is not simple. 5. (a) Let H be a proper subgroup of a ﬁnite group G . Show that if  G  does not divide [ G : H ]! then G is not simple. (b) Show that any group of order 36 cannot be simple. (You may use the result of part (a), even if you can’t prove it). 6. Let D n be the group of symmetries of a regular ngon. If n is odd, how many 2Sylow subgroups does D n have? 7. Suppose f : G → H is a group homomorphism, with H abelian, and ker f < N < G . Show that N C G . Do not use any results about commutator subgroups....
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 Fall '08
 Staff
 Group Theory, Normal subgroup, 75 minutes, scratch paper, Dr. Matthew Macauley

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