View source for Characteristic subgroup

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" />.



====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]].