Volume 12, Issue 5

Let F be a subfield of the complex numbers. An element x of a finite group G is called an F -element in G if χ ( x ) ∈ F for every character χ of G . We show that G has a unique largest normal subgroup N containing no nonidentity F -elements of G . Also, the canonical homomorphism G → G / N defines a bijection from the set of classes of F -elements of G to the set of classes of F -elements of G / N .

In this paper, it is shown that if p is an odd prime, and if P is a finite p -group, then there exists an exact sequence of abelian groups , where D ( P ) is the Dade group of P and T ( P ) is the subgroup of endo-trivial modules. Here is the group of sequences of compatible elements in the Dade groups D ( N P ( Q )/ Q ) for non-trivial subgroups Q of P . The poset is the set of elementary abelian subgroups of rank at least 2 of P , ordered by inclusion. The group is the subgroup of consisting of classes of P -invariant 1-cocycles. A key result for the proof that the above sequence is exact is a characterization of elements of 2 D ( P ) by sequences of integers, indexed by sections ( T, S ) of P such that T / S ≅ (ℤ/ p ℤ) 2 , fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p 3 and exponent p .

We propose a simplification of the definition of saturation for fusion systems over p -groups and prove the equivalence of our definition with that of Broto, Levi, and Oliver.

We consider hyperbolic triangle groups of the form T = T p 1 , p 2 , p 3 , where p 1 , p 2 , p 3 are prime numbers. Let p be a prime number and n be a positive integer. We give a necessary and sufficient condition for L 2 ( p n ) to be the image of a given hyperbolic triangle group, where L 2 ( p n ) denotes the projective special linear group PSL 2 ( p n ). It follows that, given a prime number p , there exists a unique positive integer n such that L 2 ( p n ) is the image of a given hyperbolic triangle group. Finally, given a hyperbolic triangle group T , we determine the asymptotic probability that a randomly chosen homomorphism φ : T → L 2 ( p n ) is surjective, as p n tends to infinity.

Groups of finite Morley rank generalize algebraic groups; the simple ones have even been conjectured to be algebraic. Parallel to an ambitious classification program towards this conjecture, one can try to show direct equivalents of known results on algebraic groups in the context of groups of finite Morley rank. This is done here with Steinberg's theorem on centralizers of semi-simple elements.

According to Lazard, every p -adic Lie group contains an open pro- p subgroup which is saturable. This can be regarded as the starting point of p -adic Lie theory, as one can naturally associate to every saturable pro- p group G a Lie lattice L ( G ) over the p -adic integers. Essential features of saturable pro- p groups include that they are torsion-free and p -adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p -adic analytic pro- p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p -adic analytic pro- p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p -groups of small order.

A torsion-free variety of groups is a variety generated by torsion-free groups (of course, such a non-trivial variety contains also groups which are not torsion-free). Kovács noted that possibly some torsion-free variety may be definable by a finite number of identities (as a torsion-free variety) without having any finite basis in the usual sense. However, no example of a torsion-free variety with this property was known. In the present note we give such an example. In other words, we give an example of a finitely generated verbal subgroup U of a free group F of infinite rank such that the isolator of U in F is not finitely generated as a verbal subgroup.

Let 𝔅 be a variety in which the free groups of finite rank are residually finite. We show that there exists a simple group G that is locally 𝔅-by-finite and contains a copy of the 𝔅-free group of countably infinite rank. We also prove that there exist periodic simple groups that are locally residually finite but not locally finite.

Given an integer n ≠ 0, 1, let ℬ n be the variety of n -Bell groups defined by the law [ x n , y ] = [ x, y n ], and let be the class of all groups G in which, for any infinite subsets X and Y of G , there exist x ∈ X and y ∈ Y such that [ x n , y ] = [ x, y n ]. We prove that every infinite -group G is n -Bell in the following cases: G is finitely generated and locally graded; G is locally soluble; G is locally graded and | n | or | n – 1| is equal to 2 a p b (where p is a prime, and a, b are non-negative integers). We also show that every infinite -group is a 4-Bell group.

In 1937, B. H. Neumann constructed a family of continuously many, non-isomorphic two-generator groups. We will prove that none of the infinite groups in this family are finitely presented, describe how to present them and classify their word problems. Other remarkable properties of Neumann's groups are also discussed.

We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of three free groups.