Difference between revisions of "Fully-characteristic subgroup"
From Encyclopedia of Mathematics
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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. | + | 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==== | ====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> | + | <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> | ||
Latest revision as of 18:29, 12 December 2015
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
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