Namespaces
Variants
Actions

Difference between revisions of "Characteristic subgroup"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(LaTeX)
Line 1: Line 1:
A subgroup of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217401.png" /> that is invariant under all automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217402.png" />.
+
A subgroup of a group $G$ that is invariant under all automorphisms of $G$.
  
  
  
 
====Comments====
 
====Comments====
Examples of characteristic subgroups are the [[Centre of a group|centre of a group]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217403.png" />, the [[Fitting subgroup|Fitting subgroup]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217404.png" />, the [[Commutator subgroup|commutator subgroup]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217406.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217407.png" />, the [[Frattini-subgroup(2)|Frattini subgroup]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217408.png" />, the [[Socle|socle]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c0217409.png" />, the layer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021740/c02174010.png" />, and the generalized Fitting subgroup. For a definition of the last two, cf. [[Fitting subgroup|Fitting subgroup]].
+
Examples of characteristic subgroups are the [[centre of a group]], denoted by $Z(G)$, the [[Fitting subgroup]], $F(G)$, the [[commutator subgroup]], $D(G)$, $[G,G]$ or $G'$, the [[Frattini-subgroup(2)|Frattini subgroup]], $\Phi(G)$, the [[Socle|socle]], $\mathrm{Socl}(G)$, the layer, $E(G)$, and the generalized Fitting subgroup. For a definition of the last two, cf. [[Fitting subgroup]].

Revision as of 18:06, 29 November 2014

A subgroup of a group $G$ that is invariant under all automorphisms of $G$.


Comments

Examples of characteristic subgroups are the centre of a group, denoted by $Z(G)$, the Fitting subgroup, $F(G)$, the commutator subgroup, $D(G)$, $[G,G]$ or $G'$, the Frattini subgroup, $\Phi(G)$, the socle, $\mathrm{Socl}(G)$, the layer, $E(G)$, and the generalized Fitting subgroup. For a definition of the last two, cf. Fitting subgroup.

How to Cite This Entry:
Characteristic subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_subgroup&oldid=15360
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article