We study the uniformly recurrent subgroups of simple locally finite groups.

A proper subgroup M of a finite group G is called a CCsubgroup of G if the centralizer CG(m) of every m ∈ M = M \ {1} is contained in M . Such finite groups had been partially classified by S.Williams, A. S.Kondrat’iev, N. Iiyori and H.Yamaki, M. Suzuki, W.Feit and J. G.Thompson, M.Herzog, Z.Arad, D.Chillag and others. In [6] the present authors, having taken all this work into account, classif...

A graph /is said to be G-locally primitive, where G is a subgroup of automorphisms of /-, if the stabiliser G~ of a vertex a acts primitively on the s e t / ( a ) of vertices of / adjacent to or. For a finite non-abelian simple group L and a Cayiey subset S of L, suppose that L <3 G...<Aut( L ) , and the Cayley graph /= Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simp...

Let $pounds$ be the category of locally compact abelian groups and $A,Cin pounds$. In this paper, we define component extensions of $A$ by $C$ and show that the set of all component extensions of $A$ by $C$ forms a subgroup of $Ext(C,A)$ whenever $A$ is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. ...

We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (l...