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From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: Hall subgroup and retract
View other subgroup property conjunctions | view all subgroup properties

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

Definition

A Hall retract of a finite group is a Hall subgroup that is also a retract: in other words, it possesses a normal complement. Note that the normal complement must also be a Hall subgroup, and in fact, a normal Hall subgroup.

Relation with other properties

Stronger properties

• Hall direct factor
• Sylow retract

Weaker properties

• Order-conjugate subgroup
• Order-isomorphic subgroup
• Order-automorphic subgroup
• Isomorph-automorphic subgroup
• Isomorph-conjugate subgroup
• Automorph-conjugate subgroup
• Procharacteristic subgroup
• Pronormal subgroup
• Intermediately isomorph-conjugate subgroup
• Intermediately automorph-conjugate subgroup
• Intermediately normal-to-characteristic subgroup
• Intermediately subnormal-to-normal subgroup