Automorphisms of divisible rigid groups

D. V. Ovchinnikov

Algebra i logika, Tome 53 (2014) no. 2, pp. 206-215

Résumé

A group $G$ is $m$-rigid if there exists a normal series of the form $$ G=G_1>G_2>\ldots>G_m>G_{m+1}=1 $$ in which every factor $G_i/G_{i+1}$ is an Abelian group and is torsion-free as a (right) $\mathbb Z[G/G_i]$-module. A rigid group is one that is $m$-rigid for some $m$. The specified series is determined by a given rigid group uniquely; so it consists of characteristic subgroups and is called a rigid series; the solvability length of a group is exactly $m$. A rigid group $G$ is divisible if all $G_i/G_{i+1}$ are divisible modules over $\mathbb Z[G/G_i]$. The rings $\mathbb Z[G/G_i]$ satisfy the Ore condition, and $Q(G/G_i)$ denote the corresponding (right) division rings. Thus, for a divisible rigid group $G$, the factor $G_i/G_{i+1}$ can be treated as a (right) vector space over $Q(G/G_i)$. We describe the group of all automorphisms of a divisible rigid group, and then a group of normal automorphisms. An automorphism is normal if it keeps all normal subgroups of the given group fixed.

Détail

Mots-clés : divisible rigid group
Keywords: group of automorphisms, group of normal automorphisms.