Subgroup invariant under conjugation by a generating set
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 is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
Contents
Definition
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.
Stronger properties
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 |