View source for Abnormal subgroup

A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102701.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102702.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102703.png" /> for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102704.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102705.png" /> is the subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102706.png" /> and its conjugate subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102707.png" />. As an example of an abnormal subgroup of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102708.png" /> one can take the normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a0102709.png" /> of any Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027010.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027011.png" />, and even any maximal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027012.png" /> which is not normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027013.png" />. In the theory of finite solvable groups (cf. [[Solvable group|Solvable group]]), where many important classes of subgroups are abnormal, use is made of the concept of a subabnormal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027014.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027015.png" />, which is defined by a series of subgroups

<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/a/a010/a010270/a01027016.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027017.png" /> is abnormal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027019.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR></table>



====Comments====
Nowadays, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027020.png" /> is mostly defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010270/a01027021.png" />. Instead of  "solvable"  also  "soluble groupsoluble"  is frequently used.