existence of maximal subgroups

Because every finite groupMathworldPlanetmath is a finite setMathworldPlanetmath, every chain of proper subgroupsMathworldPlanetmath of a finite group has a maximal elementMathworldPlanetmath and thus every finite group has a maximal subgroup. The same applies to maximal normal subgroups.

However, there are infinite groups, even abelianMathworldPlanetmath, with no maximal subgroups and no maximal normal subgroups. The Prüfer group


(for any prime p) is an example of an abelian group with no maximal subgroups. As the group is abelian all subgroupsMathworldPlanetmathPlanetmath are normal so it also has no maximal normal subgroups. Such groups fail to fit the hypothesisMathworldPlanetmath of the Jordan-Hölder decomposition theorem as they do not have the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath and so we cannot assign a composition seriesMathworldPlanetmathPlanetmathPlanetmath to such groups.

This stands in contrast to the categoryMathworldPlanetmath of unital rings where if one assumes Zorn’s lemma (axiom of choiceMathworldPlanetmath) then one may prove every unital ring has a maximal idealMathworldPlanetmathPlanetmath.

Title existence of maximal subgroups
Canonical name ExistenceOfMaximalSubgroups
Date of creation 2013-03-22 16:24:54
Last modified on 2013-03-22 16:24:54
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 5
Author Algeboy (12884)
Entry type Example
Classification msc 20E28
Related topic PropertyOfInfiniteSimpleGroup
Related topic JordanHolderDecomposition
Related topic EveryRingHasAMaximalIdeal