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Maximal subgroup

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

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

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 is a subgroup of satisfying .

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