Volume 16, Issue 4

Abstract. The famous Tits alternative states that a linear group either contains a non-abelian free group or is soluble-by-(locally finite). In this paper we study similar alternatives in pseudofinite groups. We show, for instance, that an ℵ 0 -saturated pseudofinite group either contains a subsemigroup of rank 2 or is nilpotent-by-(uniformly locally finite). We call a class of finite groups  G weakly of bounded rank if the radical has a bounded Prüfer rank and the index of the socle of is bounded. We show that an ℵ 0 -saturated pseudo-(finite weakly of bounded rank) group either contains a non-abelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between these kind of alternatives and amenability.

Abstract. Recently, Schlage-Puchta proved super-multiplicity of p -deficiency for normal subgroups of p -power index. We extend this result to all normal subgroups of finite index. We then use the methods of the proof to show that some groups with non-positive p -deficiency have virtually positive p -deficiency. We also compute the p -deficiency in some cases such as Fuchsian groups and study related invariants: the lower and upper absolute p -homology gradients and the p -Euler characteristic.

Abstract. For any prime number p and any positive real number we construct a finitely generated group Γ with p -gradient equal to . This construction is used to show that there exist uncountably many pairwise non-commensurable groups that are finitely generated, infinite, torsion, non-amenable, and residually- p .

Abstract. In this paper, we prove the following characterization of LFS-groups of p -type: Let G be a locally finite group and be a local system for G such that and for all : (i) is a local system for G . (ii) there exists such that (a) is block-diagonal for S for , (b) for and , (c) for , where is the number of composition factors C for on with . Then is an LFS-group of p -type and . We use this theorem to construct a general family of LFS-groups of p -type.

Abstract. Let G be a finite group and N be a normal subgroup of  G and suppose that the p -regular elements of  N have exactly two G -conjugacy class sizes. It is shown that N is solvable and that if H is a p -complement of  N , then either H is abelian or H is the product of an r -group for some prime and a central subgroup of  G .