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Difference between revisions of "Frattini-subgroup(2)"

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The [[Characteristic subgroup|characteristic subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413501.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413502.png" /> defined as the intersection of all maximal subgroups (cf. [[Maximal subgroup|Maximal subgroup]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413503.png" />, if there are any; otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413504.png" /> is its own Frattini subgroup. It was introduced by G. Frattini [[#References|[1]]]. The Frattini subgroup consists of precisely those elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413505.png" /> that can be removed from any generating system of the group containing them, that is,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413506.png" /></td> </tr></table>
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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,
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$$
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\Phi(G) = \{ x \in G : \langle M,x \rangle = G \Rightarrow \langle M \rangle = G \} \ .
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$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413507.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041350/f0413508.png" /> is nilpotent.
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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>
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<table>
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<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>
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<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>
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</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
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