A note on noninner automorphisms of order $p$ for finite $p$-groups of coclass 2

TL;DR

This paper proves the existence of noninner automorphisms of order 2 in finite 2-groups of coclass 2 and extends the result to all finite p-groups of coclass 2, showing they have noninner automorphisms of order p fixing the center.

Contribution

It establishes the existence of noninner automorphisms of order p for all finite p-groups of coclass 2, generalizing previous results.

Findings

Finite 2-groups of coclass 2 have noninner automorphisms of order 2.

All finite p-groups of coclass 2 have noninner automorphisms of order p fixing the center.

The result combines new proof with recent work by Guerboussa and Reguiat.

Abstract

In this note, the existence of noninner automorphisms of order 2 for finite 2-groups of coclass 2 is proved. Combining our result with a recent one due to Y. Guerboussa and M. Reguiat (see arXiv:1301.0085), we prove that every finite $p$-group of coclass 2 has a noninner automorphism of order $p$ leaving the center elementwise fixed.