Absolute central automorphisms that are inner.
For a finite group \(G\), the absolute centre of \(G\) is the subgroup \(L(G)\) of \(G\) left fixed elementwise by all elements of \(\Aut(G)\). By analogy with the group of central automorphisms, the authors consider the group \(\Aut_l(G)\) of those automorphisms of \(G\) which act trivially on \(G/L(G)\). Since clearly \(L(G)\) is contained in the centre \(Z(G)\) of \(G\), every element of \(\Aut_l(G)\) centralizes the group \(\text{Inn}(G)\) of the inner automorphisms of \(G\). The main result of this note is this: if \(G\) is a nonabelian autonilpotent finite \(p\)-group of class 2, then \(\Aut_l(G)=\text{Inn}(G)\) if and only if \(L(G)=Z(G)\) is cyclic.
scientific article; zbMATH DE number 6374452