On certain residually finite groups

In this paper we will give necessary and sufficient conditions under which A ⊕ B = A ⊕ C implies B and C are comparable relative to ≤ for all finitely generated projective modules A, B and C over a regular ring     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

Commutators in residually finite groups

The following result is proved. Let n be a positive integer and G a residually finite group in which every product of at most 68 commutators has order dividing n. Then G′ is locally finite.     

Centralizers in residually finite torsion groups

Let be a residually finite torsion group. We show that, if has a finite 2-subgroup whose centralizer is finite, then is locally finite. We also show that, if has no -torsion, and is a finite 2-group acting on in such a way that the centralizer is soluble, or of finite exponent, then is locally finite.

     

A footnote on residually finite groups

We give an easy proof that a finitely generated group which is residually (finite and soluble of bounded rank) is nilpotent by quasi-linear. This can be used to shorten the proofs of some recent theorems about residually finite groups.     

剩余有限minimax 可解群的几乎正则自同构

设G 是一个剩余有限的minimax 可解群, α 是G 的几乎正则自同构, 则G/[G, α] 是有限群, 并且(1) 当α^p = 1 时, G 有一个指数有限的幂零群其幂零类不超过h(p), 其中h(p) 是只与素数p 有关的函数.(2) 当α² = 1 时, G 有一个指数有限的Abel 特征子群且[G, α]′ 是有限群.关键词剩余有限minimax 可解群几乎正则自同构     

Multilinear commutators in residually finite groups

The following result is proved. Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.     

On trifactorized soluble groups of finite rank

Let the soluble-by-finite group G=AB=AC=BC be the product of two nilpotent subgroups A and B and a subgroup C. It is shown that, if G has finite abelian section rank and C is hypercentral (hypercyclic), then G is hypercentral (hypercyclic). Moreover, if G is an L ₁-group and C is nilpotent, then G is nilpotent.Dedicated to Professor Guido Zappa on the occasion of his 75th birthday     

On finite soluble groups of fixed rank

We establish the dependence of the derived length and p-length of a finite soluble group on its rank.     

Linear core conditions in residually finite groups

LetG be a residually finite or pro-finite group. We say thatG satisfies the linear core condition with constantc if all finite index (open) subgroups ofG contain a subgroup of index at mostc which is normal inG. Answering a question of L. Pyber we give a complete characterisation of finitely generated residually finite and pro-finite groups satisfying a linear core generated residually finite and pro-finite groups satisfying a linear core condition. In the case of infinitely generated groups we prove that such groups are abelian-by-finite. Research supported by the Hungarian National Research Foundation (OTKA), grant no. 16432 and F023436.     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

On residually finite graph products

We provide a simple set of sufficient conditions for the residual finiteness of a graph product of groups, which is a generalization of G. Baumslag's residual finiteness criterion for an amalgamated free product of two groups.     

On Weyl groups in minimal simple groups of finite Morley rank

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type.