# Join of Sylow subgroups

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]

## Definition

A subgroup $H$ of a finite group $G$ 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 $G$. 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 $G$.

## 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

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
join of homomorph-dominating subgroups join of subgroups of the group each of which is a homomorph-dominating subgroup of the whole group follows from Sylow implies homomorph-dominating |FULL LIST, MORE INFO
subgroup whose normal closure is homomorph-containing normal closure is a homomorph-containing subgroup (via join of homomorph-dominating subgroups) |FULL LIST, MORE INFO
join of endomorph-dominating subgroups join of subgroups of the group each of which is an endomorph-dominating subgroup of the whole group (via join of homomorph-dominating subgroups) (via join of homomorph-dominating subgroups) |FULL LIST, MORE INFO
subgroup whose normal closure is fully invariant normal closure is a fully invariant subgroup (via join of endomorph-dominating subgroups) (via join of endomorph-dominating subgroups) Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
join of automorph-conjugate subgroups join of subgroups of the group each of which is an automorph-conjugate subgroup of the whole group follows from Sylow implies automorph-conjugate |FULL LIST, MORE INFO
closure-characteristic subgroup normal closure is a characteristic subgroup (via join of automorph-conjugate subgroups) Join of automorph-conjugate subgroups, Subgroup whose normal closure is fully invariant, Subgroup whose normal closure is homomorph-containing|FULL LIST, MORE INFO
paracharacteristic subgroup contranormal in its join with any automorphic subgroup follows from Sylow implies paracharacteristic and paracharacteristicity is strongly join-closed |FULL LIST, MORE INFO
paranormal subgroup contranormal in its join with any conjugate subgroup (via paracharacteristic) |FULL LIST, MORE INFO