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

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Statement

Suppose P is a finite p-group such that the center Z(P) is a cyclic group of order p. Suppose, further, that the inner automorphism group \operatorname{Inn}(P) is a maximal subgroup in the normal closure of any p-Sylow subgroup in the automorphism group \operatorname{Aut}(P). Then, if \sigma is a p-automorphism of P (i.e., an automorphism whose order is a power of p), and Q is a group containing P as a normal subgroup, \sigma can be extended to an automorphism \sigma' of Q.

Examples

The prototypical example of this is where P is dihedral group:D8. In this case, \operatorname{Aut}(P) 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 2-automorphisms of the dihedral group (which in this case means all automorphisms) can be extended to any group containing this as a normal subgroup.

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