# 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.

However, there are infinite groups, even abelian^{}, with no maximal subgroups and
no maximal normal subgroups. The Prüfer group

$${\mathbb{Z}}_{{p}^{\mathrm{\infty}}}=\underset{\u27f5}{lim}{\mathbb{Z}}_{{p}^{i}}$$ |

(for any prime $p$) 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.

This stands in contrast to the category^{} of unital rings where if one assumes Zorn’s lemma (axiom of choice^{}) then one may prove every unital ring
has a maximal ideal^{}.

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 |