Intermediately isomorph-conjugate subgroup

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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]

Definition

Symbol-free definition

A subgroup of a group is termed intermediately isomorph-conjugate if it is isomorph-conjugate in every intermediate subgroup.

Definition with symbols

A subgroup H of a group G is termed intermediately isomorph-conjugate if given any other subgroup L of G which is isomorphic to H, there is an element x \in <H,L> such that xHx^{-1} = L.

Formalisms

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In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: isomorph-conjugate subgroup
View other properties obtained by applying the intermediately operator

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If H is intermediately isomorph-conjugate in G, then H is also intermediately isomorph-conjugate in any intermediate subgroup. This follows from the definition.

Normalizing joins

This subgroup property is normalizing join-closed: the join of two subgroups with the property, one of which normalizes the other, also has the property.
View other normalizing join-closed subgroup properties

If H, K \le G are intermediately isomorph-conjugate subgroups, and K \le N_G(H), then the join of subgroups HK is also an intermediately isomorph-conjugate subgroup. Further information: Intermediate isomorph-conjugacy is normalizing join-closed