Groups Satisfying Some Normalizer Conditions*

Some Residually Finite Groups Satisfying Laws

We give an example of a residually-p finitely generated group, that satisfies a non-trivial group law, but is not virtually solvable. Denote by Fn the free group on n generators. Recall that, given a group word m(x1, . . . , xn) ∈ Fn, a group G satisfies the law m = 1 if for every u1, . . . , un ∈ G, m(u1, . . . , un) = 1. Given a set S of group laws, the n-generator free group in the variety g...

Linear Algebraic Groups without the Normalizer Theorem

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material about general linear algebraic groups before one c...

Some Remarks on Mappings Satisfying Cyclical Contractive Conditions

In this paper, we consider, discuss and complement fixed points results for mappings satisfying cyclical contractive conditions established by W. A. Kirk et al. [W. A. Kirk, P. S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, Volume 4, No. 1, 2003, 79-89]. By using our new Lemma we get much shorter and nicer proofs of some re...

Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group an...

Some Fixed Point Theorems for Weakly Compatible Multivalued Mappings Satisfying Some General Contractive Conditions of Integral Type