# Difference between revisions of "Finite solvable group"

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==Definition== | ==Definition== | ||

− | A [[finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions: | + | A [[defining ingredient::finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions: |

− | + | # It is a [[defining ingredient::solvable group]] | |

− | + | # It is a [[defining ingredient::polycyclic group]] | |

− | + | # It has [[Sylow complement]]s for all prime divisors of the order of the group | |

− | + | # It has [[Hall subgroup]]s of all possible orders | |

+ | # All its composition factors (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order. | ||

==Relation with other properties== | ==Relation with other properties== |

## Revision as of 00:58, 15 November 2008

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

View other properties of finite groups OR View all group properties

## Definition

A finite group is termed a **finite solvable group** if it satisfies the following equivalent conditions:

- It is a solvable group
- It is a polycyclic group
- It has Sylow complements for all prime divisors of the order of the group
- It has Hall subgroups of all possible orders
- All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order.