View source for Fully-characteristic subgroup

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A subgroup of a group $G$ that is invariant with respect to all endomorphisms of $G$. The set of fully-characteristic subgroups forms a sublattice in the lattice of all subgroups. The commutator subgroup and the members of the lower central series in an arbitrary group are fully-characteristic subgroups. In addition, any [[Verbal subgroup|verbal subgroup]] of a group is fully characteristic. The converse statement is true for free groups: Any fully-characteristic subgroup is verbal.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations of groups in terms of generators and relations" , Interscience  (1966)</TD></TR></table>