Difference between revisions of "Frattini-subgroup(2)"
From Encyclopedia of Mathematics
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| − | + | The [[characteristic subgroup]] $\Phi(G)$ of a group $G$ defined as the intersection of all [[maximal subgroup]]s of $G$, if there are any; otherwise $G$ is its own Frattini subgroup. It was introduced by G. Frattini [[#References|[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 | + | 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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Frattini, "Intorno alla generazione dei gruppi di operazioni" ''Atti Accad. Lincei, Rend. (IV)'' , '''1''' (1885) pp. 281–285</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> G. Frattini, "Intorno alla generazione dei gruppi di operazioni" ''Atti Accad. Lincei, Rend. (IV)'' , '''1''' (1885) pp. 281–285</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 20:06, 18 October 2017
2020 Mathematics Subject Classification: Primary: 20D25 [MSN][ZBL]
The characteristic subgroup $\Phi(G)$ 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) |
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