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.

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