Automorphism groups of free metabelian nilpotent groups

Published in Algebra i Logika, Vol. 29, No. 6, pp. 746–751, November–December, 1990.     

Representations of metabelian groups

A knowledge of the simple representation theory of finite abelian groups is useful for understanding the representations of solvable groups, since these provide the one-dimensional representations. The representation theory of metabelian groups (those G with abelian commutator subgroup G′) would seem to be a natural next level.In this paper we shall show that these representations, too, may be simply described in several ways: they are induced from linear representations of some explicity defined subgroups; their degrees may be calculated from a knowledge of the subgroups of G; these degrees depend only on the kernel of the representation (in fact, only on the intersection of this kernel with G′). As an application of these results, we can calculate for metabelian groups a certain measure of group-commutativity studied in an earlier paper [4].     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Symmetric words in metabelian groups

An n-ary word w(x₁,…,x_n) is called n-symmetric for a group G if w(g₁,…,g_n) = w(g_{σ 1},…,g_{σ n}) for all g₁,…,g_n in G and all permu¬tations a in the symmetric group S_n. In this note we describe 2 and 3-symmetric words in free metabelian groups and metabelian groups of nilpotency class c, for arbitrary c.     

Automorphism groups of fields

We consider pairs (K,G) of an infinite field K or a formally real field K and a group G and want to find extension fields F of K with automorphism group G. If K is formally real then we also want F to be formally real and G must be right orderable. Besides showing the existence of the desired extension fields F, we are mainly interested in the question about the smallest possible size of such fields. From some combinatorial tools, like Shelah’s Black Box, we inherit jumps in cardinalities of K and F respectively. For this reason we apply different methods in constructing fields F: We use a recent theorem on realizations of group rings as endomorphism rings in the category of free modules with distinguished submodules. Fortunately this theorem remains valid without cardinal jumps. In our main result (Theorem 1) we will show that for a large class of fields the desired result holds for extension fields of equal cardinality. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag