Central Automorphisms of Semidirect Products

Hamid Mousavi ; Amir Shomali

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 3

Résumé

In this paper we describe the structure of $\text{Aut}_N^Z(G)$ for a group $G=HK$, where $K$ is a normal subgroup of $G$ and $N= H \cap K$ is $\text{Aut}^Z(G)$-invariant, in particular, if N = 1, this amounts to a description of the central automorphism group of the semi-direct product $G=K\rtimes H$. We also show that if $N\trianglelefteq G$ and $\mathcal{C}_K(H/N)=N$, then $\text{Aut}^Z_N(G)$ is a split extension. Particular if $G$ is solvable, then $\text{Aut}_N^Z(G)$ is an abelian by abelian split extension. This description of the group of central automorphisms of semidirect products is of great importance, because any solvable group has a splitting quotient.

Détail

Classification : 20D15, 20D45