Template:Random subgroup property

RANDOM SUBGROUP PROPERTY: Permutable subgroup: A subgroup that commutes with every other subgroup. May not be normal.@@@Hall subgroup: A subgroup of a finite group whose order and index are relatively prime. Sylow subgroups are a special case.@@@Automorph-conjugate subgroup: A subgroup of a group that is conjugate to any automorphic subgroup. Any characteristic subgroup is automorph-conjugate.@@@2-subnormal subgroup: A normal subgroup of a normal subgroup. Need not be normal, because normality is not transitive.@@@Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.@@@Central factor: A subgroup with the property that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.@@@Abelian normal subgroup: A subgroup that is Abelian as a group and normal as a subgroup.@@@Contranormal subgroup: A subgroup whose normal closure in the whole group is the whole group.@@@Nearly normal subgroup: A subgroup having finite index inside its normal closure.@@@Conjugacy-closed subgroup: A subgroup such that any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.@@@Local subgroup: A subgroup that occurs as the normalizer of a nontrivial solvable subgroup.