Maximal subgroup

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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 $H_0$.


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


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


[a1] M. Hall jr., "The theory of groups" , Macmillan (1959)
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Maximal subgroup. Encyclopedia of Mathematics. URL:
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