Frattini-subgroup(2)
2020 Mathematics Subject Classification: Primary: 20D25 [MSN][ZBL]
The characteristic subgroup of a group G defined as the intersection of all maximal subgroups of G, if there are any; otherwise G is its own Frattini subgroup. It was introduced by G. Frattini [1]. The Frattini subgroup consists of precisely those elements of G that can be removed from any generating system of the group containing them, that is, \Phi(G) = \{ x \in G : \langle M,x \rangle = G \Rightarrow \langle M \rangle = G \} \ .
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 G, the group \Phi(G) 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) |
Frattini-subgroup(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frattini-subgroup(2)&oldid=42118