# Difference between revisions of "Finite solvable group"

From Groupprops

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# It has [[Hall subgroup]]s of all possible orders | # 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. | # All its composition factors (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order. | ||

+ | |||

+ | ==Examples== | ||

+ | |||

+ | ===Extreme examples=== | ||

+ | |||

+ | * The [[trivial group]] is a finite solvable group. | ||

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

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===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

+ | | [[Weaker than::finite abelian group]] || finite and an [[abelian group]] || follows from [[abelian implies solvable]] || see [[solvable not implies abelian]] || {{intermediate notions short|finite solvable group|finite abelian group}} | ||

+ | |- | ||

+ | | [[Weaker than::finite nilpotent group]] || finite and a [[nilpotent group]] || follows from [[nilpotent implies solvable]] || see [[solvable not implies nilpotent]]|| {{intermediate notions short|finite solvable group|finite nilpotent group}} | ||

+ | |- | ||

+ | | [[Weaker than::finite supersolvable group]] || finite and a [[supersolvable group]] || follows from [[supersolvable implies solvable]] || see [[solvable not implies supersolvable]] || {{intermediate notions short|finite solvable group|finite supersolvable group}} | ||

+ | |} |

## Revision as of 21:29, 26 May 2011

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

## Contents

## 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.

## Examples

### Extreme examples

- The trivial group is a finite solvable group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite abelian group | finite and an abelian group | follows from abelian implies solvable | see solvable not implies abelian | Finite nilpotent group, Finite supersolvable group, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |

finite nilpotent group | finite and a nilpotent group | follows from nilpotent implies solvable | see solvable not implies nilpotent | Finite supersolvable group, Group having a Sylow tower, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |

finite supersolvable group | finite and a supersolvable group | follows from supersolvable implies solvable | see solvable not implies supersolvable | Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |