Frattini-subgroup(2)
From Encyclopedia of Mathematics
The characteristic subgroup 
of a group 
defined as the intersection of all maximal subgroups (cf. Maximal subgroup) of 
, if there are any; otherwise 
is its own Frattini subgroup. It was introduced by G. Frattini [1]. The Frattini subgroup consists of precisely those elements of 
that can be removed from any generating system of the group containing them, that is,
 |
A finite group is nilpotent if and only if its derived group is contained in its Frattini subgroup. For every finite group and every polycyclic group 
, the group 
is nilpotent.
References
| [1] | G. Frattini, "Intorno alla generazione dei gruppi di operazioni" Atti Accad. Lincei, Rend. (IV) , 1 (1885) pp. 281–285 |
| [2] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
How to Cite This Entry:
Frattini-subgroup(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frattini-subgroup(2)&oldid=42118
Frattini-subgroup(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frattini-subgroup(2)&oldid=42118
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article