Band 20, Heft 2

Let G be a finite 2-group with the property that |H:HG|≤2${|H:H_{G}|\leq 2}$ for all subgroups H of G . Then G has an abelian normal subgroup of index at most 4 in G . This result represents an affirmative answer to Question 18.56 from the current edition of the Kourovka Notebook.

The classification of p -groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph 𝒢${\mathcal{G}}$ associated with the p -groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph 𝒢∗${{\mathcal{G}}^{\ast}}$ of 𝒢${\mathcal{G}}$ associated with those p -groups of maximal class whose automorphism group orders are divisible by p-1${p-1}$. We describe the broad structure of 𝒢∗${{\mathcal{G}}^{\ast}}$ by determining its so-called skeleton. We investigate the smallest interesting case p=7${p=7}$ in more detail using computational tools, and propose an explicit version of Conjecture W for 𝒢∗${{\mathcal{G}}^{\ast}}$ for arbitrary p≥7${p\geq 7}$. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

We prove a conjecture of Boston that if p≥5${p\geq 5}$, all p -central quotients of the free group on two generators and of the free product of two cyclic groups of order p are Beauville groups. In the case of the free product, we also determine Beauville structures in p -central quotients when p=3${p=3}$. As a consequence, we give an infinite family of Beauville 3-groups, which is different from the ones that were known up to date.

Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set 𝒫${\mathcal{P}}$ of Γ, and transitively on the lines of Γ. We show that the primitive action of G on 𝒫${\mathcal{P}}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on 𝒫${\mathcal{P}}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.

The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V⁢(ℤ⁢G)${\mathrm{V}(\mathbb{Z}G)}$, the group of normalized units of the integral group ring of the finite group G , it must be isomorphic to a subgroup of G . The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However, the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p -groups of rank 2. We give a positive solution to the Subgroup Isomorphism Problem for C4×C2${C_{4}\times C_{2}}$. Moreover, we prove that if the Sylow 2-subgroup of G is a dihedral group, any 2-subgroup of V⁢(ℤ⁢G)${\mathrm{V}(\mathbb{Z}G)}$ is isomorphic to a subgroup of G .

L -algebras arise in algebraic logic, in the theory of one-sided lattice-ordered groups, and in connection with set-theoretic solutions of the quantum Yang–Baxter equation. They apply in several ways to Garside groups. For example, the set of primitive elements, the set of simple elements, and the negative cone of a Garside group are all L -algebras. Picantin’s iterated crossed product decomposition of Garside groups can be reformulated and extended in terms of L -algebras. It is proved that the structure group of an L -algebra, introduced in connection with the “logic” of ℓ${\ell}$-groups, is torsion-free. This applies to the left group of fractions of not necessarily noetherian, Garside-like monoids which need not embed into their ambient group.

Near-isomorphism is known as the right concept for classification theorems in the theory of almost completely decomposable groups. As a natural generalization the authors extended in [6] the notion of near-isomorphism to Abelian groups of arbitrary rank. In this article we investigate non-isomorphic direct decompositions of a class of infinite rank torsion-free abelian groups which were defined in [4] as special epimorphic images of so-called almost rigid groups. A complete classification of such decompositions up to near-isomorphism is given.

We study the group IAut⁡(A)${\operatorname{IAut}(A)}$ generated by the inertial automorphisms of an abelian group A , that is, automorphisms γ${{\gamma}}$ with the property that each subgroup H of A has finite index in the subgroup generated by H and H⁢γ${H{\gamma}}$. Clearly, IAut⁡(A)${\operatorname{IAut}(A)}$ contains the group FAut⁡(A)${\operatorname{FAut}(A)}$ of finitary automorphisms of A , which is known to be locally finite. In a previous paper, we showed that IAut⁡(A)${\operatorname{IAut}(A)}$ is (locally finite)-by-abelian. In this paper, we show that IAut⁡(A)${\operatorname{IAut}(A)}$ is also metabelian-by-(locally finite). More precisely, IAut⁡(A)${\operatorname{IAut}(A)}$ has a normal subgroup Γ${{\Gamma}}$ such that IAut⁡(A)/Γ${\operatorname{IAut}(A)/{\Gamma}}$ is locally finite and the derived subgroup Γ′${{\Gamma}^{\prime}}$ is an abelian periodic subgroup all of whose subgroups are normal in Γ${{\Gamma}}$. In the case when A is periodic, IAut⁡(A)${\operatorname{IAut}(A)}$ turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of IAut⁡(A)${\operatorname{IAut}(A)}$.

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over ℤ${\mathbb{Z}}$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over ℤ${\mathbb{Z}}$ must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.

A subgroup H≤G${H\leq G}$ is commensurated if |H:H∩gHg-1|<∞$|H:H\cap gHg^{-1}|<\infty$ for all g∈G${g\in G}$. We show that a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group and many other branch groups of independent interest is either finite or of finite index.

Let G , H be groups and let κ${{\kappa}}$ be a cardinal. A bijection f:G→H${f:G\to H}$ is called an asymorphism if, for any X∈[G]<κ${X\in[G]^{<{\kappa}}}$, Y∈[H]<κ${Y\in[H]^{<{\kappa}}}$, there exist X′∈[G]<κ${X^{\prime}\in[G]^{<{\kappa}}}$, Y′∈[H]<κ${Y^{\prime}\in[H]^{<{\kappa}}}$ such that for all x∈G${x\in G}$ and y∈H${y\in H}$, we have f⁢(X⁢x)⊆Y′⁢f⁢(x)${f(Xx)\subseteq Y^{\prime}f(x)}$, f-1⁢(Y⁢y)⊆X′⁢f-1⁢(y)${f^{-1}(Yy)\subseteq X^{\prime}f^{-1}(y)}$. For a set S , [S]<κ${[S]^{<{\kappa}}}$ denotes the set {S′⊆S:|S′|<κ}${\{S^{\prime}\subseteq S:|S^{\prime}|<{\kappa}\}}$. Let κ${{\kappa}}$ and γ be cardinals such that ℵ0<κ≤γ${\aleph_{0}<{\kappa}\leq\gamma}$. We prove that any two Abelian groups of cardinality γ are κ${{\kappa}}$-asymorphic, but the free group of rank γ is not κ${{\kappa}}$-asymorphic to an Abelian group provided that either κ<γ${{\kappa}<\gamma}$ or κ=γ${{\kappa}=\gamma}$ and κ${{\kappa}}$ is a singular cardinal. It is known [9] that if γ=κ${\gamma={\kappa}}$ and κ${{\kappa}}$ is regular, then any two groups of cardinality κ${{\kappa}}$ are κ${{\kappa}}$-asymorphic.