Unformatted text preview: H E K and K E G , it does not follow in general that H E G . Recall that an automorphism of a group G is an isomorphism from G to itself. We say that a subgroup H ≤ G is characteristic in G , and write H char G , if for all automorphisms φ of G , φ ( H ) = H . 1. * (a). Show that if H char G , then H E G . (b). Let H < K < G , where H char K and K E G . Then H E G . (c). Show that if K is a cyclic subgroup of G and K E G , then every subgroup H of K satisﬁes H E G . 2. * Let f,g : R → R be the functions deﬁned by the formulas f ( x ) =x and g ( x ) = x +1. Let G = h f,g i be the subgroup of the group S R of all permutations of the set R which is generated by f and g . Prove carefully that G ∼ = h a,b  b 2 = e,ba = a1 b i . (This group is called the inﬁnite dihedral group , A ∞ .) 1...
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 Fall '08
 Bunch,J
 Math, Group Theory, Normal subgroup, Foote, Dummit

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