A Construction of Smooth Travel Groupoids on Finite Graphs

A travel groupoid is an algebraic system related with graphs. In this paper, we give an algorithm to construct smooth travel groupoids for any finite graph. This algorithm gives an answer of Nebeský’s question, “Does there exist a connected graph G such that G has no smooth travel groupoid?”, in finite cases.     

On the diameter of the intersection graph of a finite simple group

Let G be a finite group. The intersection graph Δ_G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected.     

Frattini and related subgroups of mapping class groups

Let Γ_g,b denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let Φ_f(G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φ_f(G) is nilpotent when G is a finitely generated subgroup of Γ_g,b, in this paper we compute Φ_f(G) for certain subgroups of Γ_g,b. In particular, we answer Ivanov’s question in the affirmative for these subgroups of Γg,b.     

Transitivity degrees of countable groups and acylindrical hyperbolicity

We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here, by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.     

Residually finite algorithmically finite groups,their subgroups and direct products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group G, thus answering a question of Myasnikov and Osin. The group G here is “strongly infinite” and “strongly algorithmically finite,” which means that G contains an infinite Abelian normal subgroup and all finite Cartesian powers of G are algorithmically finite (i.e., for any n, there is no algorithm writing out infinitely many pairwise distinct elements of the group G ⁿ ). We also formulate several open questions concerning this topic.     

On locally finite Cpp-groups

A group G is called a Cpp-group for a prime number p, if G has elements of order p and the centralizer of every non-trivial p-element of G is a pgroup. In this paper we prove that the only infinite locally finite simple groups that are Cpp-groups are isomorphic either to PSL(2,K) or, if p = 2, to Sz(K), with K a suitable algebraic field over GF(p). Using this fact, we also give some structure theorems for infinite locally finite Cpp-groups.     

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space Aⁿ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.     

Finite groups with <Emphasis Type="Italic">SS</Emphasis>-supplement

Let G be a finite group. A subgroup H of G is said to be SS-quasinormal in G if there is a subgroup K such that \(G=HK\) and \(HS=SH\), for all \(S\in \) Syl(K), where Syl(K) denotes the collection of all Sylow subgroups of K. A subgroup H of G is said to be SS-supplemented in G if there is a subgroup K such that \(G=HK\) and \(H\cap K\) is SS-quasinormal in G. In this paper, we investigate the SS-supplemented subgroups and strengthen a result of Skiba which gives a positive answer to an open question of Shemetkov.     

QUANDLE VARIETIES,GENERALIZED SYMMETRIC SPACES,AND <Emphasis Type="Italic">φ</Emphasis>-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ?. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry.Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ?φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.     

On a question of Külshammer for representations of finite groups in reductive groups

Let G be a simple algebraic group of type G₂ over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ₂ and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ₂ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H¹(Γ, V) → H¹(Γ_p, V) induced by restriction of 1-cocycles has an infinite fibre.     

On <Emphasis Type="Italic">G</Emphasis>-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

Realizability of Two-dimensional Linear Groups over Rings of Integers of Algebraic Number Fields

Given the ring of integers O _K of an algebraic number field K, for which natural numbers n there exists a finite group G???GL(n, O _K ) such that O _K G, the O _K -span of G, coincides with M(n, O _K ), the ring of (n?×?n)-matrices over O _K ? The answer is known if n is an odd prime. In this paper we study the case n?=?2; in the cases when the answer is positive for n?=?2, for n?=?2m there is also a finite group G???GL(2m, O _K ) such that O _K G?=?M(2m, O _K ).     

Coleman automorphisms of holomorphs of completely reducible and almost simple groups

Let H be the holomorph of a finite group G. It is proved that every Coleman automorphism of H is inner whenever G is either completely reducible or almost simple; in particular, this is the case when G is either characteristically simple or simple. As an application, we obtain the normalizer the conjecture holds for integral group rings of holomorphs of such groups in question.     

Decomposition of Tensor Products Involving a Steinberg Module

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G _r or a finite Chevalley group \(G(\mathbb {F}_q)\). In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G _r or \(G(\mathbb {F}_q)\). Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 2²ⁿ⁺¹ and either q ? 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F⁴(q), where q = 2 ⁿ and either q⁴ + 1 or q⁴ ? q² + 1 is a prime number.     

On the torus cobordant cohomology spheres

Let G be a compact Lie group. In 1960, P A Smith asked the following question: “Is it true that for any smooth action of G on a homotopy sphere with exactly two fixed points, the tangent G-modules at these two points are isomorphic?” A result due to Atiyah and Bott proves that the answer is ‘yes’ for ? _p and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on S ⁿ which are c-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in ?-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Characterization of the alternating groups by their order and one conjugacy class length

Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2.