Automorphisms of divisible rigid groups

D. Ovchinnikov

A group G is said to be rigid if it has a normal series \[G=G_1>G_2>\ldots >G_n>G_{n+1}=1\] in which each factor \(G_i/G_{i+1}\) is an abelian group that is torsion free as a \(\mathbb{Z}[G/G_i]\)-module. Rigid groups are introduced in [1] in connection with algebraic geometry over groups. Important examples of rigid groups are free solvable groups. \(G\) is divisible [2] if all \(G_i/G_{i+1}\) are divisible modules. Any \(\mathbb{Z}[G/G_i]\) is a (right) Ore domain, and we denote by \(Q(G_i/G_{i+1})\) the (right) Ore skew field of fractions of \(\mathbb{Z}[G/G_i]\). Therefore, for a divisible rigid group \(G\), the quotient \(G_i/G_{i+1}\) can be viewed as a vector space over \(Q(G_i/G_{i+1})\). \(G\) is splittable if it splits into a semidirect product \(A_1A_2\ldots A_n\) of abelian groups \(A_i\cong G_i/G_{i+1}\). A splittable divisible rigid group is uniquely defined by the dimensions of \(A_i\) over \(Q_i=Q(A_1\ldots A_{i-1})\) (denoted by \(\alpha_i\)). It is proved that every divisible rigid group is splittable [3] and any rigid group is embedable into some divisible rigid group [2].

We study the groups of automorphisms of divisible rigid groups.

Theorem 1. For a divisible rigid group \(G\), the group \(Aut(G)\cong MN\) is a semidirect product of \(M=GL_{\alpha_1}(Q_1) \cdot \ldots \cdot GL_{\alpha_n}(Q_n)\) and a normal subgroup \(N=\{ \phi_g \ | \ g\in A_2\ldots A_n\} \cong A_2\ldots A_n\) which is a group of corresponding inner automorphisms.

The second result concerns the normal automorphisms of \(G\). Recall that \(\phi\in Aut(G)\) is normal if, for any normal subgroup \(H\) of \(G\), \(\phi(H)=H\).

Theorem 2. The group of normal automorphisms of \(G\) is equal to \(Inn(G) \times \mathbb{Z}_2\), where \(\mathbb{Z}_2=\langle e_0\rangle\) and \(e_0 : a_i\to a_i\) for \(i \lt n\), \(e_0: a_n\to a_n^{-1}\), \(a_i\in A_i\).

References

1. A. Myasnikov, N. Romanovskiy, Krull dimension of solvable groups, J. Algebra, 324, N10 (2010), 2814–2831.
2. N. S. Romanovskiy, Divisible rigid groups, Algebra Logic, 47, N6 (2008), 426–434.
3. A. Myasnikov, N. Romanovskiy, Logical aspects of divisible rigid groups. (to appear)

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