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
Canonical name	ExampleOfFullyInvariantSubgroup
Date of creation	2013-03-22 16:04:41
Last modified on	2013-03-22 16:04:41
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	5
Author	juanman (12619)
Entry type	Example
Classification	msc 20D99
Related topic	FullyInvariantSubgroup