Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
From Groupprops
Contents
Statement
Suppose is a group with center
, inner automorphism group
, and automorphism group
. Suppose the following two statements are true:
- Every automorphism of
is a center-fixing automorphism: it fixes every element of the center
.
-
is a maximal subgroup of
.
Then, every automorphism of is a normal-extensible automorphism: whenever
is a normal subgroup of some group
and
is an automorphism of
,
extends to an automorphism
of
.
Related facts
- Centerless and maximal in automorphism group implies every automorphism is normal-extensible
- Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible
- Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner
Facts used
- Center-fixing implies central factor-extensible: If
is a center-fixing automorphism of a central factor
of a group
,
extends to an automorphism of
.
- Equivalence of definitions of central factor
Proof
Given: A group such that every automorphism of
fixes every element of
, and
is a maximal subgroup of
.
To prove: For any automorphism of
, and any group
containing
as a normal subgroup,
extends to an automorphism of
.
Proof: Let be the homomorphism given by the action of
on
by conjugation. This map exists because
is normal in
.
- The image of
is either
or
: The image of
is a subgroup of
. It contains
, because
equals
. Since
is a maximal subgroup of
, the image is either
or
.
- If the image is
, then every automorphism of
extends to an inner automorphism of
, and we are done. (We say in this case that
is a normal fully normalized subgroup of
).
- If the image is
, then
is a central factor of
, i.e.,
(see fact (2)). Thus, by fact (1), any automorphism
that fixes the center of
extends to an automorphism of
. Since we assumed that every automorphism fixes the center of
, we obtain that every automorphism of
extends to an automorphism of
.