The Small Index Property for Free Nilpotent Groups

Abstract

Let F be a relatively free algebra of infinite rank κ. We say that F has the small index property if any subgroup of Γ = Aut(F ) of index at most κ contains the pointwise stabilizer Γ (U ) of a subset U of F of cardinality less than κ. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.

Automorphisms of Aut(A),Aa Free Abelian Group

“…In the present paper, a sequel to the paper [10], we prove, extending the results by Hua and Reiner, that all automorphisms of the group Aut A where A is an infinitely generated free abelian group are inner: Aut Aut A = Inn Aut A . To prove this we use the small index property for the group A, which was established in [10].…”

“…As was proved in [9], is generated by all -moietous automorphisms and, as a corollary, is perfect. As a corollary of Theorem 3.1 of [10], one gets the following result on the generation of pointwise stabilizers of . An involution ∈ is called extremal if the fixed-point subgroup Fix ≤ A of is of corank one and there exists a unimodular element x ∈ A such that…”

“…Following [10], we call a subgroup of having index at most rank A a subgroup of small index. As a corollary of the main result of [10], every subgroup of of small index contains the pointwise stabilizer U of a subset U of A of cardinality < rank A .…”

“…To prove this we use the small index property for the group A, which was established in [10]. Alternatively, the main result of the present paper can be obtained with the use of the general theory of isomorphisms of linear groups over division rings developed by O. O'Meara [7] (it is required to use at some step the result on the perfectness of the group Aut A from the author's paper [9]).…”

“…The paper is organized as follows. In the first section, using results about the definability of conjugacy classes of involutions in = Aut A [8] and the small index property for A [10], we show that given any unimodular element x ∈ A, the stabilizer…”