This preview shows page 1. Sign up to view the full content.
Unformatted text preview: : G X be a bijection. Prove that there is an operation on X which makes X into a group such that : G X is an isomorphism. Solution. Absent. 2.67 (i) Prove that the composite of homomorphisms is itself a homomor-phism. Solution. If f : G H and g : H K are homomorphisms, then ( g f )( ab ) = g ( f ( ab )) = g ( f a f b ) = g ( f a ) g ( f b ) = ( g f )( a )( g f )( b ). (ii) Prove that the inverse of an isomorphism is an isomorphism. Solution. If f is an isomorphism, then f 1 ( x ) = a if and only if x = f ( a ) . Hence, if f 1 ( y ) = b , then f 1 ( xy ) = ab (since f ( ab ) = f ( a ) f ( b ) = xy ), and so f 1 ( xy ) = ab = f 1 ( x ) f 1 ( y ) ....
View Full Document
- Fall '11