Volume 20, Issue 6

A remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally FP2{\mathrm{FP}_{2}}) subgroups. We prove the analogous result for relatively hyperbolic groups of Bredon cohomological dimension 2 with respect to the family of parabolic subgroups. A class of groups where our result applies consists of C′⁢(1/6){C^{\prime}(1/6)} small cancellation products. The proof relies on an algebraic approach to relative homological Dehn functions, and a characterization of relative hyperbolicity in the framework of finiteness properties over Bredon modules and homological Isoperimetric inequalities.

It is proved that every group of the form G=A⁢B{G=AB} with two subgroups A and B each of which is either abelian or has a quasicyclic subgroup of index 2 is soluble of derived length at most 3. In particular, if A is abelian and B is a locally quaternion group, this gives a positive answer to Question 18.95 of the “Kourovka notebook” posed by A. I. Sozutov.

A group homomorphism e:H→G{e:H\to G} is a cellular cover of G if for every homomorphism φ:H→G{\varphi:H\to G} there is a unique homomorphism φ¯:H→H{\bar{\varphi}:H\to H} such that φ¯⁢e=φ{\bar{\varphi}e=\varphi}. Group localizations are defined dually. The main purpose of this paper is to establish 2ℵ0{2^{\aleph_{0}}} varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Göbel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 2ℵ0{2^{\aleph_{0}}} varieties not closed under localizations. Finally, the existence of 2ℵ0{2^{\aleph_{0}}} varieties of groups neither closed under cellular covers nor under localizations is presented as well.

Let G be a group containing a nilpotent normal subgroup N with central series {Nj}{\{N_{j}\}} such that each Nj/Nj+1{N_{j}/N_{j+1}} is an 𝔽{\mathbb{F}}-vector space over a field 𝔽{\mathbb{F}} and the action of G on Nj/Nj+1{N_{j}/N_{j+1}} induced by the conjugation action is 𝔽{\mathbb{F}}-linear. For k∈ℕ{k\in\mathbb{N}} we describe a necessary and sufficient condition for all elements from any coset xN , x∈G{x\in G}, to admit k th roots in G , in terms of the action of x on the quotients Nj/Nj+1{N_{j}/N_{j+1}}. This yields in particular a condition for surjectivity of the power maps, generalising various results known in special cases. For 𝔽{\mathbb{F}}-algebraic groups we also characterise the property in terms of centralisers of elements. For a class of Lie groups, it is shown that surjectivity of the k th power map, k∈ℕ{k\in\mathbb{N}}, implies the same for the restriction of the map to the solvable radical of the group. The results are applied in particular to the study of exponentiality of Lie groups.

In this paper, we study the fibers of “automorphic word maps”, a certain generalization of word maps, on finite groups and on nonabelian finite simple groups in particular. As an application, we derive a structural restriction on finite groups G where, for some fixed nonempty reduced word w in d variables and some fixed ρ∈(0,1]{\rho\in(0,1]}, the word map wG{w_{G}} on G has a fiber of size at least ρ⁢|G|d{\rho|G|^{d}}: No sufficiently large alternating group and no (classical) simple group of Lie type of sufficiently high rank can occur as a composition factor of such a group G .

Let G be a finite group having a factorisation G=A⁢B{G=AB} into subgroups A and B with B cyclic and A∩B=1A\cap B=1, and let b be a generator of B . The associated skew-morphism is the bijective mapping f:A→A{f:A\to A} well-defined by the equality b⁢a⁢B=f⁢(a)⁢B{baB=f(a)B}, where a∈A{a\in A}. In this paper, we shall classify all skew-morphisms of cyclic p -groups, where p is an odd prime.

Let p be a prime. We show that other than a few exceptions, alternating groups will have p -blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p -subgroup P and G/P{G/P} is nilpotent if and only if φ⁢(1)2{\varphi(1)^{2}} divides |G:ker(φ)|{|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G .

For a prime number p and a finite group G , we consider the set of p -Brauer characters of G taking values in some field 𝔽{\mathbb{F}} (called 𝔽{\mathbb{F}}-Brauer characters) and the set of conjugacy classes of p -regular elements of G whose value at every Brauer character lies in 𝔽{\mathbb{F}} (called 𝔽{\mathbb{F}}-conjugacy classes). We show that these sets for G and for G/N{G/N} are closely related, for certain normal subgroups N of G . This is a p -modular version of a result of Isaacs and Navarro in characteristic 0 (see [3]).

We say that a finite almost simple G with socle S is admissible (with respect to the spectrum) if G and S have the same sets of orders of elements. Let L be a finite simple linear or unitary group of dimension at least three over a field of odd characteristic. We describe admissible almost simple groups with socle L . Also we calculate the orders of elements of the coset L⁢τ{L\tau}, where τ is the inverse-transpose automorphism of L .