# 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=11533

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