Automorphism that preserves conjugacy classes for a generating set
From Groupprops
This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Contents
Definition
Suppose is a group and
is an automorphism of
.We say that
is an automorphism that preserves conjugacy classes for a generating set if the folllowing equivalent conditions hold:
- There is a generating set
for
such that, for all
,
is in the conjugacy class of
, i.e., there exists
such that
. Note that
depends on
.
- There is a symmetric subset
of
that is a generating set of
and such that, for all
,
is in the conjugacy class of
, i.e., there exists
such that
. Note that
depends on
.
- There is a normal subset
of
that is a generating set of
and such that, for all
,
is in the conjugacy class of
, i.e., there exists
such that
. Note that
depends on
.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
inner automorphism | an automorphism for which there exists a single element ![]() ![]() |
(obvious) | (via class-preserving automorphism) | |FULL LIST, MORE INFO |
locally inner automorphism | looks like an inner automorphism in its effect on any finite subset | (obvious) | (via class-preserving automorphism) | |FULL LIST, MORE INFO |
class-preserving automorphism | sends every element to within its conjugacy class | (obvious) | preserves conjugacy classes for a generating set not implies class-preserving | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
IA-automorphism | sends every element to within its coset of the derived subgroup, or equivalently, induces the identity automorphism on the abelianization | preserves conjugacy classes for a generating set implies IA | IA-automorphism|automorphism that preserves conjugacy classes for a generating set]] |