Join of Sylow subgroups

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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)	|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)	|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)		|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
polycharacteristic subgroup		(via paracharacteristic)		|FULL LIST, MORE INFO
polynormal subgroup		(via paranoral, also via polycharacteristic)		|FULL LIST, MORE INFO