# Fully-characteristic subgroup

From Encyclopedia of Mathematics

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 of a group is fully characteristic. The converse statement is true for free groups: Any fully-characteristic subgroup is verbal.

#### References

[1] | W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations of groups in terms of generators and relations" , Interscience (1966) |

**How to Cite This Entry:**

Fully-characteristic subgroup.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fully-characteristic_subgroup&oldid=36893

This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article