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
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.