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Difference between revisions of "Maximal subgroup"

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A proper [[Subgroup|subgroup]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629901.png" /> which is not contained in any other proper subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629902.png" />, that is, a maximal element in the set of proper subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629903.png" /> ordered by inclusion. There exist groups without maximal subgroups, for example, a [[Group-of-type-p^infinity|group of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629904.png" />]].
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A proper [[Subgroup|subgroup]] of a group $G$ which is not contained in any other proper subgroup of $G$, that is, a maximal element in the set of proper subgroups of $G$ ordered by inclusion. There exist groups without maximal subgroups, for example, a [[Group-of-type-p^infinity|group of type $p^\infty$]].
  
A generalization of the concept of a maximal subgroup is that of a subgroup maximal with respect to some property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629905.png" />, i.e. a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629906.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629907.png" /> with the property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629908.png" /> and such that no other proper subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m0629909.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299010.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299011.png" /> and contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299012.png" />.
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A generalization of the concept of a maximal subgroup is that of a subgroup maximal with respect to some property $\sigma$, i.e. a subgroup $H_0$ of $G$ with the property $\sigma$ and such that no other proper subgroup $H$ of $G$ has $\sigma$ and contains $H_0$.
  
 
====References====
 
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====Comments====
 
====Comments====
A proper subgroup of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299013.png" /> is a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299015.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062990/m06299016.png" />.
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A proper subgroup of a group $G$ is a subgroup $H$ of $G$ satisfying $H\neq G$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</TD></TR></table>

Latest revision as of 22:17, 17 July 2014

A proper subgroup of a group $G$ which is not contained in any other proper subgroup of $G$, that is, a maximal element in the set of proper subgroups of $G$ ordered by inclusion. There exist groups without maximal subgroups, for example, a group of type $p^\infty$.

A generalization of the concept of a maximal subgroup is that of a subgroup maximal with respect to some property $\sigma$, i.e. a subgroup $H_0$ of $G$ with the property $\sigma$ and such that no other proper subgroup $H$ of $G$ has $\sigma$ and contains $H_0$.

References

[1] M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)


Comments

A proper subgroup of a group $G$ is a subgroup $H$ of $G$ satisfying $H\neq G$.

References

[a1] M. Hall jr., "The theory of groups" , Macmillan (1959)
How to Cite This Entry:
Maximal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_subgroup&oldid=32492
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article