Open main menu

Groupprops β

Subgroup for which any join of conjugates is a join of finitely many conjugates

Definition

Suppose H is a subgroup of a group G. We say that H is a subgroup for which any join of conjugates is a join of finitely many conjugates if, for any subset S of G, there is a finite subset T of G such that:

\langle \bigcup_{s \in S} sHs^{-1} \rangle = \langle \bigcup_{t \in T} tHt^{-1}\rangle

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup equals all of its conjugates (obvious) Almost normal subgroup, Nearly normal subgroup|FULL LIST, MORE INFO
Subgroup of finite index has finite index in the whole group Almost normal subgroup, Nearly normal subgroup|FULL LIST, MORE INFO
Subgroup of finite group the whole group is a finite group |FULL LIST, MORE INFO
Nearly normal subgroup has finite index in its normal closure |FULL LIST, MORE INFO
Almost normal subgroup has finitely many conjugate subgroups |FULL LIST, MORE INFO