Volume 27, Issue 2

In this paper, we consider several basic facts of Schur covers of the symmetric groups and braid groups. In particular, we give explicit presentations of Schur covers of braid groups.

We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear CAT⁢(0)\mathrm{CAT}(0), C⁢(6)C(6) graphical small cancellation, and a systolic one. We then use these structures to establish various properties of the fundamental groups of these complexes, such as biautomaticity and the Tits Alternative. We isolate an easily checkable condition implying hyperbolicity of the fundamental groups, and we construct some non-hyperbolic examples. We also briefly discuss a parallel theory of C⁢(4)C(4)-T⁢(4)T(4) graphical complexes of groups and outline their basic properties.

By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.

Let Fq\mathbb{F}_{q} be the finite field with 𝑞 elements and consider the 𝑛-dimensional Fq\mathbb{F}_{q}-vector space V=FqnV=\mathbb{F}_{q}^{n}. In this paper, we define a closure operator on the subgroup lattice of the group G=PGL⁢(V)G=\mathrm{PGL}(V). Let 𝜇 denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups 𝐻 of 𝐺 for which μ⁢(H,G)≠0\mu(H,G)\neq 0. Moreover, we establish a polynomial bound on the number c⁢(m)c(m) of closed subgroups 𝐻 of index 𝑚 in 𝐺 for which the lattice of 𝐻-invariant subspaces of 𝑉 is isomorphic to a product of chains. This bound depends only on 𝑚 and not on the choice of 𝑛 and 𝑞. It is achieved by considering a similar closure operator for the subgroup lattice of GL⁢(V)\mathrm{GL}(V) and the same results proven for this group.

The Engel graph of a finite group 𝐺 is a directed graph encoding the pairs of elements in 𝐺 satisfying some Engel word. Recent work of Lucchini and the third author shows that, except for a few well-understood cases, the Engel graphs of almost simple groups are strongly connected. In this paper, we give a refinement to this analysis.

It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.

We consider the quotient group T⁢(G)T(G) of the multiple holomorph by the holomorph of a finite 𝑝-group 𝐺 of class two for an odd prime 𝑝. By work of the first-named author, we know that T⁢(G)T(G) contains a cyclic subgroup of order pr−1⁢(p−1)p^{r-1}(p-1), where prp^{r} is the exponent of the quotient of 𝐺 by its center. In this paper, we shall exhibit examples of 𝐺 (with r=1r=1) such that T⁢(G)T(G) has order exactly p−1p-1, which is as small as possible.

A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that G/N′G/N^{\prime} is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that G/N′G/N^{\prime} belongs to 𝔛. Examples have been given in [F. de Giovanni, M. Trombetti and B. A. F. Wehfritz, Hall classes of groups, to appear] to show that finite-by-𝔛 groups do not form a Hall class for many natural choices of the Hall class 𝔛. Although these examples are often linear, our aim here is to prove that the situation is much better within certain natural subclasses of the universe of linear groups.

We describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order p2p^{2}. We also describe all projective representations of Heisenberg groups with entries from the rings Z/p2⁢Z\mathbb{Z}/p^{2}\mathbb{Z} and Fp⁢[t]/(t2)\mathbb{F}_{p}[t]/(t^{2}) for odd primes 𝑝 and obtain a classification of their degenerate and non-degenerate 2-cocycles.