On Periodic Groups Saturated with Finite Frobenius Groups

A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

ON ALGORITHMIC SOLVABILITY OF THE PROBLEM OF GENERALIZED CONJUGACY OF SUBGROUPS IN COXETER GROUPS WITH A TREE STRUCTURE

The main algorithmic problems of combinatorial group theory posed by M. Den and G. Titze at the beginning of the twentieth century are the problems of word, word conjugacy and of group isomorphism. However, these problems, as follows from the results of P.S. Novikov and S.I. Adyan, turned out to be unsolvable in the class of finitely defined groups. Therefore, algorithmic problems began to be considered in specific classes of groups. The word conjugacy problem allows for two generalizations. On the one hand, we consider the problem of conjugacy of subgroups, that is, the problem of constructing an algorithm that allows for any two finitely generated subgroups to determine whether they are conjugate or not. On the other hand, the problem of generalized conjugacy of words is posed, that is, the problem of constructing an algorithm that allows for any two finite sets of words to determine whether they are conjugated or not. Combining both of these generalizations into one, we obtain the problem of generalized conjugacy of subgroups. Coxeter groups were introduced in the 30s of the last century, and the problems of equality and conjugacy of words are algorithmically solvable in them. To solve other algorithmic problems, various subclasses are distinguished. This is partly due to the unsolvability in Coxeter groups of another important problem – the problem of occurrence, that is, the problem of the existence of an algorithm that allows for any word and any finitely generated subgroup of a certain group to determine whether this word belongs to this subgroup or not. The paper proves the algorithmic solvability of the problem of generalized conjugacy of subgroups in Coxeter groups with a tree structure.

Towards Constructing Fully Homomorphic Encryption without Ciphertext Noise from Group Theory

In CRYPTO 2008, 1 year earlier than Gentry’s pioneering “bootstrapping” technique for the first fully homomorphic encryption (FHE) scheme, Ostrovsky and Skeith III had suggested a completely different approach towards achieving FHE. They showed that the $$\mathsf {NAND}$$ operator can be realized in some non-commutative groups; consequently, homomorphically encrypting the elements of the group will yield an FHE scheme, without ciphertext noise to be bootstrapped. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism $$\pi :\widetilde{G} \rightarrow G$$ between finite groups for which bit operators are realized in G and the elements of the kernel of $$\pi $$ are indistinguishable from the general elements of $$\widetilde{G}$$. Secondly, we propose new methodologies to realize bit operators in some groups G. Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the “linearity” of the construction. Then we propose an idea to avoid such “linearity” by using combinatorial group theory. Concretely realizing FHE schemes based on our proposed framework is left as a future research topic.

A Non-Adaptive Combinatorial Group Testing Strategy to Facilitate Healthcare Worker Screening During the Severe Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) Outbreak

Background: Regular SARS-CoV-2 testing of healthcare workers (HCWs) has been proposed to prevent healthcare facilities becoming persistent reservoirs of infectivity. Using monoplex testing, widespread screening would be prohibitively expensive, and throughput may not meet demand. We propose a non-adaptive combinatorial (NAC) group-testing strategy to increase throughput and facilitate rapid turnaround via a single round of testing. Methods: NAC matrices were constructed for sample sizes of 700, 350 and 250 with replicates of 2, 4 and 5, respectively. Matrix performance was tested by simulation under different SARS-CoV-2 prevalence scenarios of 0.1-10%, with each simulation ran for 10,000 iterations. Outcomes included the proportions of re-tests required and the proportion of true negatives identified. NAC matrices were compared to Dorfman Sequential (DS) approaches. A web application (www.samplepooling.com) was designed to decode results. Findings: NAC matrices performed well at low prevalence levels with an average number of 585 tests saved per assay in the n=700 matrix at a 1% prevalence. As prevalence increased, matrix performance deteriorated with n=250 most tolerant. In simulations of low to medium (0.1%-3%) prevalence levels all NAC matrices were superior, as measured by fewer repeated tests required, to the DS approaches. At very high prevalence levels (10%) the DS matrix was marginally superior, however both group testing approaches performed poorly at high prevalence levels. Interpretation: This testing strategy maximises the proportion of samples resolved after a single round of testing, allowing prompt return of results to staff members. Using the methodology described here, laboratories can adapt their testing scheme based on required throughput and the current population prevalence, facilitating a data-driven testing strategy.