existence of maximal subgroups
Because every finite group is a finite set, every chain of proper subgroups of a finite group has a maximal element and thus every finite group has a maximal subgroup. The same applies to maximal normal subgroups.
(for any prime ) is an example of an abelian group with no maximal subgroups. As the group is abelian all subgroups are normal so it also has no maximal normal subgroups. Such groups fail to fit the hypothesis of the Jordan-Hölder decomposition theorem as they do not have the ascending chain condition and so we cannot assign a composition series to such groups.
|Title||existence of maximal subgroups|
|Date of creation||2013-03-22 16:24:54|
|Last modified on||2013-03-22 16:24:54|
|Last modified by||Algeboy (12884)|