prove that if there is monomorphism from group A to group B and monomorphism from group B to group A than these two groups are isomorphic

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- Jun 9th 2012, 02:38 AMtarasovisomorphismbetween groups
prove that if there is monomorphism from group A to group B and monomorphism from group B to group A than these two groups are isomorphic

- Jun 9th 2012, 03:33 AMDevenoRe: isomorphismbetween groups
this is false. let F2 be the free group on 2 letters (say a,b) and let F3 be the free group on 3 letters (say x,y,z).

then a→x, b→y is a monomorphism of F2 into F3, and x→a^{2}, y→ab and z→b^{2}is a monomorphism of F3 into F2, but these groups are NOT isomorphic.