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]
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 .
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
|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|