# example of fully invariant subgroup

The derived subgroup $[G,G]$ is a fully invariant subgroup because if $f$ is an endomorphism of $G$, then for each word of commutators $[a_{1},b_{1}][a_{2},b_{2}]\cdots[a_{m},b_{m}]$, we have

 $f([a_{1},b_{1}][a_{2},b_{2}]\cdots[a_{m},b_{m}])=[fa_{1},fb_{1}][fa_{2},fb_{2}% ]\cdots[fa_{m},fb_{m}]\in[G,G]$

i.e. the homomorphic image of a word of commutators is a word of commutators.

Title example of fully invariant subgroup ExampleOfFullyInvariantSubgroup 2013-03-22 16:04:41 2013-03-22 16:04:41 juanman (12619) juanman (12619) 5 juanman (12619) Example msc 20D99 FullyInvariantSubgroup