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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 of G , ( 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 satises H E G . 2. * Let f,g : R R be the functions dened 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 = a-1 b i . (This group is called the innite dihedral group , A .) 1...
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This note was uploaded on 12/11/2011 for the course MATH 200a taught by Professor Bunch,j during the Fall '08 term at UCSD.
- Fall '08