Volume 26, Issue 6

In the context of Higman embeddings of recursive groups into finitely presented groups, we suggest an approach, termed the 𝐻-machine, which for certain wide classes of groups allows constructive Higman embeddings of recursive groups into finitely presented groups. The approach is based on Higman operations, and it explicitly constructs some specific recursively enumerable sets of integer sequences arising during the embeddings. Specific auxiliary operations are introduced to make the work with Higman operations a simpler and more intuitive procedure. Also, an automated mechanism of constructive embeddings of countable groups into 2-generator groups preserving certain “patterns” is mentioned.

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and show that if the number of generators of the cyclic presentation is greater than two, then the corresponding group is large. Generalising and extending earlier results of the authors, we describe the star graphs of orientable and non-orientable cyclic presentations and classify the cyclic presentations whose star graph components are pairwise isomorphic incidence graphs of generalised polygons, thus classifying the so-called ( m , k , ν ) (m,k,\nu) -special cyclic presentations.

For a primary abelian group 𝐺, Chekhlov and Danchev (2015) defined three variations of Kaplansky’s notion of full transitivity by restricting one’s attention to the sub group , the sub ring and the unitary subring of the endomorphism ring of 𝐺 generated by the collection of all commutator endomorphisms. They posed the problem of describing exactly which totally projective groups exhibit these forms of full transitivity. This problem, and some closely related questions, are completely answered using the Ulm function of 𝐺.

Let 𝐺 be a linear algebraic group defined over a finite field F q \mathbb{F}_{q} . We present several connections between the isogenies of 𝐺 and the finite groups of rational points ( G ⁢ ( F q n ) ) n ≥ 1 (G(\mathbb{F}_{\smash{q^{n}}}))_{n\geq 1} . We show that an isogeny ϕ : G ′ → G \phi\colon G^{\prime}\to G over F q \mathbb{F}_{q} gives rise to a subgroup of fixed index in G ⁢ ( F q n ) G(\mathbb{F}_{\smash{q^{n}}}) for infinitely many 𝑛. Conversely, we show that if 𝐺 is reductive, the existence of a subgroup H n H_{n} of fixed index 𝑘 for infinitely many 𝑛 implies the existence of an isogeny of order 𝑘. In particular, we show that the infinite sequence H n H_{n} is covered by a finite number of isogenies. This result applies to classical groups GL m \mathrm{GL}_{m} , SL m \mathrm{SL}_{m} , SO m \mathrm{SO}_{m} , SU m \mathrm{SU}_{m} , Sp 2 ⁢ m \mathrm{Sp}_{2m} and can be extended to non-reductive groups if 𝑘 is prime to the characteristic. As a special case, we see that if 𝐺 is simply connected, the minimal indices of proper subgroups of G ⁢ ( F q n ) G(\mathbb{F}_{\smash{q^{n}}}) diverge to infinity. Similar results are investigated regarding the sequence ( G ⁢ ( F p ) ) p (G(\mathbb{F}_{p}))_{p} by varying the characteristic 𝑝.

We discuss representations of finite groups having a common central 𝑝-subgroup 𝑍, where 𝑝 is a prime number. For the principal 𝑝-blocks, we give a method of constructing a relative 𝑍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann’s results on stable equivalences of Morita type to relative 𝑍-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analog of that given by Ishioka and the first author.

The prime graph Δ ⁢ ( G ) \Delta(G) of a finite group 𝐺 is a graph whose vertex set is the set of prime factors of the degrees of all irreducible complex characters of 𝐺, and two distinct primes 𝑝 and 𝑞 are joined by an edge if the product p ⁢ q pq divides some character degree of 𝐺. In 2014, Tong-Viet [H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra 397 (2014), 18–31] proposed the following conjecture. Let 𝐺 be a group and let k ≥ 5 k\geq 5 be odd. If the prime graph Δ ⁢ ( G ) \Delta(G) is 𝑘-regular, then Δ ⁢ ( G ) \Delta(G) is a complete graph of order k + 1 k+1 . In this paper, we show that if the prime graph Δ ⁢ ( G ) \Delta(G) of a finite nonsolvable group 𝐺 is 5-regular, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} or possibly the graph depicted in the first figure below. Moreover, if 𝐺 is an almost simple group, then Δ ⁢ ( G ) \Delta(G) is isomorphic to the complete graph K 6 K_{6} .

The weak commutativity group χ ⁢ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We present new bounds for the exponent of χ ⁢ ( G ) \chi(G) and its sections, when 𝐺 is a finite 𝑝-group.

In a 2021 paper, Conder et al. constructed two infinite families of chiral 4-polytopes of type { 4 , 4 , 4 } \{4,4,4\} with solvable automorphism groups. Here we present a general construction for chiral polytopes of type { 4 , 4 , … , 4 } \{4,4,\dots,4\} with rank 4, 5 and 6, which are obtained as Boolean covers of the unique tight regular polytope of the same type.