Elliptic subgroup
From Groupprops
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
Contents
Definition
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 of a group
is termed elliptic if for any subgroup
of
,
form an elliptic pair of subgroups. In other words, there exists an
such that:
where each is written 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 |