Permutably complemented Hall subgroup

From Groupprops
Jump to: navigation, search
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: permutably complemented subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup H of a finite group G is termed a permutably complemented Hall subgroup if it satisfies the following two conditions:

  • H is a Hall subgroup of G: the order and index of H are relatively prime.
  • H is a permutably complemented subgroup of G: there exists a subgroup K of G such that H \cap K is trivial and HK = G (in other words, H and K are permutable complements). Note that K also must be a Hall subgroup of G: if H is \pi-Hall, K is \pi'-Hall.

Relation with other properties

Stronger properties

Weaker properties