Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
From Groupprops
Contents
Statement
Suppose is a Conjugacy-separable group (?): in other words, given any two elements
of
that are not conjugate, there exists a normal subgroup of finite index
in
such that the images of
in
are not conjugate in
.
Suppose, further, that the set of primes that divide the order of some non-identity element of
is finite.
Then, if is an Extensible automorphism (?) of
,
is a Class-preserving automorphism (?) of
.
Related facts
- Finite-extensible implies class-preserving
- Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving
- Conjugacy-separable implies every semidirectly extensible automorphism is class-preserving
- Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
Facts used
- Semidirectly extensible implies linearly pushforwardable for representation over prime field
- Every finite group admits infinitely many sufficiently large prime fields
- Sufficiently large implies splitting, Splitting implies character-separating, Character-separating implies class-separating
Proof
Given: A conjugacy-separable group with only finitely many primes
dividing the orders of elements of
. An extensible automorphism
of
. Two elements
of
that are not conjugate.
To prove: cannot send
to
.
Proof:
- There exists a normal subgroup
of finite index in
such that the images of
and
are not conjugate in
: This follows from the definition of conjugacy-separable.
- Let
. Then, there exists a prime
such that the field of
elements is sufficiently large for
, and such that
does not divide the order of any element of
: By fact (2), there are infinitely many sufficiently large prime fields for
, i.e., there are infinitely many primes
for which the corresponding prime field is sufficiently large for
. Since there are only finitely many prime divisors of orders of elements, we can find a prime
not among any of these divisors such that the corresponding prime field is sufficiently large.
- The field of
elements is a class-separating field for
. In particular, there is a finite-dimensional linear representation
of
over this field such that
and
are not conjugate: This follows from fact (3).
-
is linearly pushforwardable over the prime field with
elements, for the
chosen above. In particular, if
, then
and
are conjugate for any representation
over this field: Let
be a representation of
over this field. Let
be the corresponding vector space and
the semidirect product for the action. Since
does not divide the order of any element of
,
is the set of elements of
of order dividing
. In particular,
is characteristic in
, and thus, if
extends to an automorphism
of
, then
also restricts to an automorphism
of
. Fact (1) thus yields that
, so
is linearly pushforwardable over the field of
elements. In particular, if
, then
, so
is conjugate to
by
.
- Let
be the linear representation chosen in step (3), and let
where
is the quotient map. Then,
and
are not conjugate in the general linear group
. However, by step (4), we have that
is linearly pushforwardable, so if
, then
and
are conjugate. This gives a contradiction, so we cannot have
, and we are done.