Volume 10, Issue 6

We prove that if G is a group with a bound on the lengths of finite subgroups and G has finite Bredon cohomological dimension, then this dimension is bounded by the sum of the previous bound and the projective dimension of a certain ℤ G -module.

We give a simple combinatorial criterion for a group that, when satisfied, implies that the group cannot be strongly relatively hyperbolic. The criterion applies to several classes of groups, such as surface mapping class groups, Torelli groups, and automorphism and outer automorphism groups of free groups.

Let G be a finitely generated residually finite group and A a finitely generated normal subgroup. Then G and A are naturally embedded in their respective profinite completions Ĝ and  . The inclusion A → G induces a morphism (continuous homomorphism) ι :  → Ĝ , and the image of ι is the closure  of A in Ĝ . If A happens to be a direct factor of G , i.e. G = A × B for some normal subgroup B , then the profinite topology on G induces the profinite topology on A , so that the morphism ι is injective and may be used to identify  with Ā ; and Ĝ = Ā × . In [D. Goldstein and R. M. Guralnick. The direct product theorem for profinite groups. J. Group Theory 9 (2006), 317–322, Question 3.1], Goldstein and Guralnick ask whether the converse holds: if  is a direct factor of Ĝ , does it follow that A is a direct factor of G ? (We take the hypothesis to mean: ‘ι is injective and Ā is a direct factor of Ĝ ’.) We show that the answer is ‘no’ in general, but ‘yes’ in a suitably restricted category, namely the virtually polycyclic groups.

Let 𝒞 be a class of finite groups closed under taking subgroups, quotients and extensions. We use a pro-𝒞 analogue of the HNN-construction, to show that every virtually torsion-free pro-𝒞 group G can be embedded in a pro-𝒞 group E such that every finite subgroup of E is, up to conjugation, contained in a finite subgroup of E isomorphic to the quotient G/F , where F is an open torsion-free normal subgroup of G . Moreover the virtual cohomological dimensions of G and E coincide. As a by-product we provide a structure theorem for cyclic p -extensions of free pro- p groups.

If G is a group acting on a set Ω, and α, β ∈ Ω, the digraph whose vertex set is Ω and whose arc set is the orbit (α, β) G is called an orbital digraph of G . Each orbit of the stabilizer G α acting on Ω is called a suborbit of G . A digraph is locally finite if each vertex is adjacent to at most finitely many other vertices. A locally finite digraph Γ has more than one end if there exists a finite set of vertices X such that the induced digraph Γ\ X contains at least two infinite connected components; if there exists such a set containing precisely one element, then Γ has connectivity one . In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivityone orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterized in a previous paper by the author.

Let G denote a finite group of permutations of a finite set Ω. Given an orbit function ω : Ω → ℝ and a class function ƒ : G → ℝ and their extensions to subsets of Ω and G , consider the quantity m ω,ƒ , which is the maximal value of ω (Γ)ƒ( C G (Γ)) over all nontrivial subsets Γ of Ω. Denote by ℳ the set of non-empty subsets Γ of Ω which satisfy m (Γ)ƒ( C G (Γ) = m ω,ƒ . Our main result, Theorem 1, states that if ℳ contains a unique maximal or minimal element Δ with respect to inclusion, then Δ g = Δ for each g ∈ G and C G (Δ) is a normal subgroup of G . This result may have potential future applications. As an example of an application of Theorem 1 we prove that if either G is a simple group or it is transitive on Ω, T is a normal subset of G not containing 1, Θ is a G -invariant subset of Ω and Γ ⊆ Ω, then (see Theorem 6).

We prove that if is a non-empty Fitting class, π = π() and G is a group such that every chief factor of G / G is an -group, then G has at least one -injector and any two -injectors are conjugate in G . This result is used to resolve an open problem and generalize some known results.

Let G be a finite p -group such that x Z( G ) ⊆ x G for all x ∈ G – Z( G ), where x G denotes the conjugacy class of x in G . Then | G | divides |Aut( G )|, where Aut( G ) is the automorphism group of G .