Unformatted text preview: G . 2.4.5. For any subgroup A of a group G , and g 2 G , show that gAg ± 1 is a subgroup of G . 2.4.6. Show that every subgroup of an abelian group is normal. 2.4.7. Let ' W G ±! H be a homomorphism of groups with kernel N . For a;x 2 G , show that '.a/ D '.x/ , a ± 1 x 2 N , aN D xN: Here aN denotes f an W n 2 N g . 2.4.8. Let ' W G ±! H be a homomorphism of G onto H . If A is a normal subgroup of G , show that '.A/ is a normal subgroup of H . 2.4.9. Deﬁne a map ² from D n to C 2 D f˙ 1 g by ².±/ D 1 if ± does not interchange top and bottom of the ngon, and ².±/ D ± 1 if ± does interchange top and bottom of the ngon. Show that ² is a homomorphism. The following exercises examine an important homomorphism from S n to C 2 D f˙ 1 g (for any n )....
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 Fall '08
 EVERAGE
 Algebra, Normal subgroup, Symmetry group, Homomorphism, homomorphism property

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