On commuting automorphisms and central automorphisms of finite 2-groups of almost maximal class

Abstract

Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, $\mathcal {A}(G)=\lbrace \alpha \in Aut(G) :x\alpha (x)=\alpha (x)x~~for~ all~ x\in G\rbrace$ is equal to the group of all central automorphisms, $Aut_{c}(G)$, except only for five ones. Also, we determined the structure of $Aut_{c}(G)$ and $\mathcal {A}(G)$ for these five groups. Using these results, in this paper, we find the structure of $\mathcal {A}(G)=Aut_{c}(G)$ for the remaining 2-groups of almost maximal class. Also, we prove the following results:

• (1) We characterise the upper central series of these groups.

• (2) We find the necessary and sufficient conditions on 2-groups of almost maximal class in order that $\mathcal {A}(Inn(G))$ to be equal to $Aut_{c}(Inn(G))$ or $Aut_{c}(G)$ to be equal to Z(Inn(G)), that is, $Aut_{c}(G)$, is as small as possible.

• (3) Also, we show that if G is a group of almost maximal class and order $2^{n}$, $n\ge 7$ in which $2\le d(G^{\prime })\le 3$, then Aut(G) is an $\mathcal {A}$-group.