A generalization of Sims’ conjecture for finite primitive groups and two point stabilizers in primitive groups

In this paper, we propose a refinement of Sims’ conjecture concerning the cardinality of the point stabilizers in finite primitive groups, and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group 𝐺 on Ω and two distinct points α , β ∈ Ω \alpha,\beta\in\Omega with G α ⁢ β ⊴ G α G_{\alpha\beta}\unlhd G_{\alpha} and G α ⁢ β ≠ 1 G_{\alpha\beta}\neq 1 , where G α G_{\alpha} is the stabilizer of 𝛼 in 𝐺 and G α ⁢ β G_{\alpha\beta} is the stabilizer of 𝛼 and 𝛽 in 𝐺. In particular, this example gives an answer to a question raised independently by Cameron and by Fomin in the Kourovka Notebook.

A proof of a conjecture by Jabara on groups in which all involutions are odd transpositions

In this paper, we prove that if 𝐺 is a group generated by elements of order two with the property that the product of any two such elements has order 1, 2, 3 or 5 with all possibilities occurring, then G ≃ A 5 G\simeq A_{5} or G ≃ PSU ⁢ ( 3 , 4 ) G\simeq\mathrm{PSU}(3,4) . This provides an affirmative answer to Problem 19.36 in the Kourovka notebook.

Compact groups with many elements of bounded order

Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1–7] for {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup.

Characteristic classes of involutions in nonsolvable groups

Let {G,D_{0},D_{1}} be finite groups such that {D_{0}\trianglelefteq D_{1}} are groups of automorphisms of G that contain the inner automorphisms of G. Assume that {D_{1}/D_{0}} has a normal 2-complement and that {D_{1}} acts fixed-point-freely on the set of {D_{0}} -conjugacy classes of involutions of G (i.e., {C_{D_{1}}(a)D_{0}<D_{1}} for every involution {a\in G} ). We prove that G is solvable. We also construct a nonsolvable finite group that possesses no characteristic conjugacy class of nontrivial cyclic subgroups. This shows that an assumption on the structure of {D_{1}/D_{0}} above must be made in order to guarantee the solvability of G and also yields a negative answer to Problem 3.51 in the Kourovka notebook, posed by A. I. Saksonov in 1969.

Elementary nets (carpets) over a discrete valuation ring

Elementary net (carpet) σ = ( σij) is called closed (admissible) if the elementary net (carpet) group E(σ ) does not contain a new elementary transvections. The work is related to the question of V. M. Levchuk 15.46 from the Kourovka notebook( closedness (admissibility) of the elementary net (carpet)over a field). Let R be a discrete valuation ring, K be the field of fractions of R, σ = (σ ij) be an elementary net of order n over R, ω = (ωij) be a derivative net for , and ωij is ideals of the ring R. It is proved that if K is a field of odd characteristic, then for the closedness (admissibility) of the net , the closedness (admissibility) of each pair (σ ij ; σ ji) is sufficient for all i ̸= j.

On the solvability of regular subgroups in the holomorph of a finite solvable group

We exhibit infinitely many natural numbers [Formula: see text] for which there exists at least one insolvable group of order [Formula: see text], and yet the holomorph of every solvable group of order [Formula: see text] has no insolvable regular subgroup. We also solve Problem 19.90(d) in the Kourovka notebook.

ON A QUESTION OF A. SOZUTOV

In the paper an answer to a problem posed by A.I. Sozutov in the Kourovka Notebook is given. The solution is based on some modification of the method that was proposed for constructing a non-abelian analogue of the additive group of rational numbers, i.e. a group whose center is an infinite cyclic group and any two non-trivial subgroups of which have a non-trivial intersection.