Pronormal Hall subgroup
From Groupprops
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: pronormal subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties
Contents
Definition
Definition with symbols
A subgroup of a finite group
is termed a pronormal Hall subgroup if it satisfies the following two conditions:
-
is a Hall subgroup of
: The order and index of
in
are relatively prime.
-
is a pronormal subgroup of
: If
is a conjugate subgroup to
in
, then
are conjugate subgroups inside
.
Relation with other properties
Stronger properties
- Sylow subgroup
- Intermediately isomorph-conjugate Hall subgroup: Also related:
Weaker properties
- Hall subgroup: For full proof, refer: Hall not implies pronormal
- Hall WNSCDIN-subgroup