# Join of Sylow subgroups

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

A subgroup of a finite group is termed a **join of Sylow subgroups** or **join of Hall subgroups** if it satisfies the following equivalent conditions:

- It can be expressed as a join of Sylow subgroups of . There are no restrictions on the prime numbers we can use for the Sylow subgroups: we could use a join of Sylow subgroups all for the same prime, or all for different primes, or with multiple primes, some of which are used multiple times.
- It can be expressed as a join of Hall subgroups of .

## Formalisms

### In terms of the join operator

This property is obtained by applying the join operator to the property: Sylow subgroup

View other properties obtained by applying the join operator

## Relation with other properties

### Stronger properties

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

Sylow subgroup | subgroup of maximal prime power order in finite group | (obvious) | (obvious) | Hall subgroup|FULL LIST, MORE INFO |

Hall subgroup | subgroup whose order and index are relatively prime | Hall implies join of Sylow subgroups | join of Sylow subgroups not implies Hall | |FULL LIST, MORE INFO |

Sylow-closure | normal closure of a Sylow subgroup | join of all the conjugates of that Sylow subgroup | |FULL LIST, MORE INFO |