In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

Alfred Schild has established conditions that Lorentz transformationsmap world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.

This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in $\text{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

Faultline configurations in top management teams are of vital importance in predicting firm outcomes. Grounded in faultline theory, we hypothesise the positive effects of faultlines through the dual routes of coordination and information processing under conditions of various subgroup configuration types. Second-hand data from publicly traded Chinese information technology firms are used to test our hypotheses. The results demonstrate that TMT faultline strength is positively related to a firm’s short-term performance only when both the number and the balance of subgroups are high and is positively related to a firm’s innovation activities only when the number of subgroups is high and the balance of subgroups is low. This study contributes to faultline theory by enriching the connotation of faultlines with the configurational perspective and advancing the debate on the effects of team faultlines as we reveal the benefits of TMT faultlines.

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

We show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

We define a notion of genericity for arbitrary subgroups of groups interpretable in a simple theory, and show that a type generic for such a group is generic for the minimal hyperdefinable supergroup (the definable hull). In particular, at least one generic type of the definable hull is finitely satisfiable in the original subgroup. If the subgroup is a subfield. then the additive and the multiplicative definable hull both have bounded index in the smallest hyperdefinable superfield.

The eight species of the melanogaster species subgroup of the genus Drosophila were compared to each other with respect to: (a) the electrophoretic mobility of the major chorion proteins under SDS–PAGE; (b) the overall eggshell ultrastructure. The phylogenetic tree obtained using the first criterion showed that the eight species are divided into three clusters that contain the species of the melanogaster complex (D. melanogaster, D. sechellia, D. mauritiana and D. simulans), the two species of the yakuba complex (D. yakuba and D. teissieri) and the erecta complex species (D. erecta and D. orena). This clustering is in agreement to the generally accepted phylogeny of the subgroup based on exclusive criteria. The eggshell ultrastructure of the species at the main body of the egg revealed that the phylogenetically closely related species exhibit similar eggshell structure. We conclude that the criteria applied in this study could be eventually used in groups with unknown phylogeny to trace the phylogenetic relationships of the species.