Difference between revisions of "Maximal subgroup"
From Encyclopedia of Mathematics
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| − | A proper [[Subgroup|subgroup]] of a group | + | {{TEX|done}} |
| + | 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 | + | 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==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | A proper subgroup of a group | + | 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
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