Subgroup invariant under conjugation by a generating set
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]
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A subgroup of a group is termed a subgroup invariant under conjugation by a generating set if there is a generating set for such that conjugation by any element of sends to a subset of itself, i.e., for all .
Note that the definition is in terms of the existence of a generating set, and does not say that the condition must hold for every generating set.
Relation with other properties
Collapse to normality
- For a finite subgroup, this condition is equivalent to being normal.
- For a subgroup of finite index, this condition is equivalent to being normal.
- The condition is equivalent to normality in a slender group as well as in an Artinian group -- and in particular in a periodic group.
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|Normal subgroup||invariant under conjugation by all elements||Conjugate-comparable subgroup|FULL LIST, MORE INFO|
|Conjugate-comparable subgroup||comparable with all its conjugate subgroups||conjugate-comparable implies invariant under conjugation by a generating set||invariant under conjugation by a generating set not implies conjugate-comparable|||FULL LIST, MORE INFO|
|Subgroup invariant under a generating set of the automorphism group||invariant under a generating set of the automorphism group||invariant under a generating set of the automorphism group implies invariant under conjugation by a generating set||any finite example of normal not implies characteristic|