View source for Abnormal subgroup

A subgroup $A$ of a group $G$ such that $g\in\langle A,A^g\rangle$ for any element $g\in G$. Here $\langle A,A^g\rangle$ is the subgroup generated by $A$ and its conjugate subgroup $A^g=gAg^{-1}$. As an example of an abnormal subgroup of a finite group $G$ one can take the normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) $N_G(P)$ of any Sylow $p$-subgroup $P\subset G$, and even any maximal subgroup $N\subset G$ which is not normal in $G$. 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 $A$ of a group $G$, which is defined by a series of subgroups

$$A=A_0\subset A_1\subset\ldots\subset A_n=G,$$

where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$.

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



====Comments====
Nowadays, $A^g$ is mostly defined as $A^g=g^{-1}Ag$. Instead of  "solvable"  also  "soluble"  is frequently used.