A homomorphism between groups is trivial if its image is the

A homomorphism between groups is trivial if its image is the neutral element. Prove that there are no non-trivial homomorphism from any finite group to any free group.

Solution

let s be aset with respect to which free group is defined.

Now as any finite group is finitely generated by fundamental theorem of finite group

By theorem

Let G be finite group generated by s={aiiI} where I is finite.

By theorem

let g be a finite group generated by s (say)and let G be any group. If ai for iI are any elements in G, not necessarily distinct, then there is at most one homomorphism :GG such that (ai)=ai. If G is free on A, then there is exactly one such homomorphism.

so here set s is nonempty with more than one element so for different a_{i   we get diff images} . so homomorphism is nontrivial .