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| − | 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 | + | 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 |
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| − | <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>
| + | $$A=A_0\subset A_1\subset\ldots\subset A_n=G,$$ |
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| − | 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" />. | + | where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$. |
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| | ====References==== | | ====References==== |
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| | ====Comments==== | | ====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. | + | Nowadays, $A^g$ is mostly defined as $A^g=g^{-1}Ag$. Instead of "solvable" also "soluble" is frequently used. |
Revision as of 10:05, 27 August 2014
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
$$A=A_0\subset A_1\subset\ldots\subset A_n=G,$$
where $A_i$ is abnormal in $A_{i+1}$, $i=0,\dots,n-1$.
References
| [1] | B. Huppert, "Endliche Gruppen" , 1 , Springer (1967) |
Nowadays, $A^g$ is mostly defined as $A^g=g^{-1}Ag$. Instead of "solvable" also "soluble" is frequently used.
How to Cite This Entry:
Abnormal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abnormal_subgroup&oldid=12360
This article was adapted from an original article by A.I. Kostrikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article