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]


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.


BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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


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