Difference between revisions of "Abnormal subgroup"
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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 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 | ||
| − | $$A=A_0\subset A_1\subset\ | + | $$A=A_0\subset A_1\subset\dotsb\subset A_n=G,$$ |
where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$. | where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$. | ||
Latest revision as of 12:05, 14 February 2020
A subgroup $A$ of a group $G$ such that $g\in\langle A,A^g\rangle$ for any element $g\in G$. Here $\langle A,A^g\rangle$ is the subgroup generated by $A$ and its conjugate subgroup $A^g=gAg^{-1}$. As an example of an abnormal subgroup of a finite group $G$ one can take the normalizer (cf. Normalizer of a subset) $N_G(P)$ of any Sylow $p$-subgroup $P\subset G$, and even any maximal subgroup $N\subset G$ which is not normal in $G$. In the theory of finite solvable groups (cf. Solvable group), where many important classes of subgroups are abnormal, use is made of the concept of a subabnormal subgroup $A$ of a group $G$, which is defined by a series of subgroups
$$A=A_0\subset A_1\subset\dotsb\subset A_n=G,$$
where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$.
References
| [1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |
Comments
Nowadays, $A^g$ is mostly defined as $A^g=g^{-1}Ag$. Instead of "solvable" also "soluble" is frequently used.
Abnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abnormal_subgroup&oldid=33149