Automorphisms of Divisible Rigid Groups

作者：Ovchinnikov D V^*

来源：Algebra and Logic, 2014, 53(2): 133-139.

DOI：10.1007/s10469-014-9277-6

• divisible rigid group
• group of automorphisms
• group of normal automorphisms

A group G is m-rigid if there exists a normal series of the form G = G(1) > G(2) > . . . > Gm > G(m+1) = 1 in which every factor G(i)/G(i+1) is an Abelian group and is torsion-free as a (right) 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 Z[G/G(i)]. The rings 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.

• 出版日期2014-5