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Difference between revisions of "Fully-characteristic subgroup"

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A subgroup of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041920/f0419201.png" /> that is invariant with respect to all endomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041920/f0419202.png" />. 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.
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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====
 
====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>

Revision as of 06:12, 1 May 2014

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=32019
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article