Semidirectly extensible implies linearly pushforwardable for representation over prime field
Statement
Suppose is a prime field (i.e., either a field of prime order or the field of rational numbers), and
is a group. Suppose
is a finite-dimensional vector space over
, and
be a linear representation of
. Let
with respect to the induced action of
on
.
Suppose, further, that is an automorphism of
that can be extended to an automorphism
of
such that
also restricts to an automorphism
of
. Then,
where
is conjugation by
in
.
Note that we need the field to be a prime field in order that is equal to the automorphism group of
as a group.
Related facts
Applications
- Finite-extensible implies class-preserving
- Hall-semidirectly extensible implies class-preserving
- Finite solvable-extensible implies class-preserving
- Conjugacy-separable with only finitely many prime divisors of orders of elements implies every extensible automorphism is class-preserving
- Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
Facts used
- Automorphism group equals general linear group for vector space over prime field
- Automorphism group action lemma: Suppose
is a group, and
are subgroups such that
. Suppose
is an automorphism of
such that the restriction of
to
gives an automorphism
of
, and such that
also restricts to an automorphism of
, say
. Consider the map:
that sends an element to the automorphism of
induced by conjugation by
(note that this is an automorphism since
). Then, we have:
where denotes conjugation by
in the group
.
Proof
Given: A group , a homomorphism
for a finite-dimensional vector space
over a prime field
.
is an automorphism of
that extends to an automorphism
of
, such that
also restricts to an automorphism
of
.
To prove: .
Proof: Since is a prime field,
is the whole automorphism group of
by fact (1) (in general, it is a proper subgroup). Thus, the element
, which is a group automorphism of
, is actually in
. Thus, fact (2), setting
, gives the desired result.