Elliptic 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]
This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties


Symbol-free definition

A subgroup of a group is termed elliptic if it forms an elliptic pair of subgroups with every subgroup of the group.

Definition with symbols

A subgroup H of a group G is termed elliptic if for any subgroup K of G, (H,K) form an elliptic pair of subgroups. In other words, there exists an n such that:

\langle H,K \rangle = (HK)^n := HKHKHK \ldots HK

where each is written n times.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup Permutable subgroup|FULL LIST, MORE INFO
permutable subgroup permutes with every subgroup |FULL LIST, MORE INFO
subgroup of finite index Subgroup of finite double coset index|FULL LIST, MORE INFO