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
Statement
Suppose is a finite
-group such that the center
is a cyclic group of order
. Suppose, further, that the inner automorphism group
is a maximal subgroup in the normal closure of any
-Sylow subgroup in the automorphism group
. Then, if
is a
-automorphism of
(i.e., an automorphism whose order is a power of
), and
is a group containing
as a normal subgroup,
can be extended to an automorphism
of
.
Examples
The prototypical example of this is where is dihedral group:D8. In this case,
is also a dihedral group of order eight, with the inner automorphism group a Klein four-subgroup. Clearly, all the conditions for the statement are satisfied, and hence, all the
-automorphisms of the dihedral group (which in this case means all automorphisms) can be extended to any group containing this as a normal subgroup.