Volume 27, Issue 3

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander–Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids and permutations, Ann. Inst. Fourier (Grenoble) 70 (2020), 3, 1341–1362], we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group VT n \mathrm{VT}_{n} on n ≥ 2 n\geq 2 strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group KT n \mathrm{KT}_{n} inside VT n \mathrm{VT}_{n} . As a by-product, it also follows that the twin group T n \mathrm{T}_{n} embeds inside the virtual twin group VT n \mathrm{VT}_{n} , which is an analogue of a similar result for braid groups.

A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.

We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct product if all groups are finitely generated torsion-free nilpotent and have a directly indecomposable rational Malcev completion. We show this by determining the complete automorphism group of the direct product.

Consider the Macdonald groups G ⁢ ( α ) = ⟨ A , B ∣ A [ A , B ] = A α , B [ B , A ] = B α ⟩ G(\alpha)=\langle A,B\mid A^{[A,B]}=A^{\alpha},\,B^{[B,A]}=B^{\alpha}\rangle , α ∈ Z \alpha\in\mathbb{Z} . We fill a gap in Macdonald’s proof that G ⁢ ( α ) G(\alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of G ⁢ ( α ) G(\alpha) .

Let 𝜎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 𝜎-subnormal subgroup is modular.

We introduce the 𝑞-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under 𝑞-isoclinism. We prove that the 𝑞-Schur multiplier is invariant under 𝑞-exterior isoclinism, and as an easy consequence, we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if 𝐺 and 𝐻 are 𝑝-groups with G / Z ∧ ⁢ ( G ) ≅ H / Z ∧ ⁢ ( H ) G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H) , then the cardinalities of the minimal number of generators of 𝐺 and 𝐻 are the same. Moreover, we prove some structural results about non-abelian 𝑞-tensor square of groups.

To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picent gr ⁢ ( A ) \mathrm{Picent}^{\mathrm{gr}}(A) of isomorphism classes of invertible 𝐺-graded ( A , A ) (A,A) -bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a Picent \mathrm{Picent} version of the Beattie–del Río exact sequence, involving Dade’s group G ⁢ [ B ] G[B] , which relates Picent gr ⁢ ( A ) \mathrm{Picent}^{\mathrm{gr}}(A) , Picent ⁢ ( B ) \mathrm{Picent}(B) , and group cohomology.