Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

On primitive permutation groups with nontrivial global stabilizers

In the paper the distinguishing number D(G) of an arbitrary finite primitive permutation group G is determined. As a consequence, the distinguishing number D(Г) of an arbitrary finite graph Г with a vertex-primitive group of automorphisms is found.     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and \({[C_G(u), C_G(v),\dots,C_G(v)]=1}\) for any \({u,v\in B{\setminus}\{1\}}\), where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every \({b \in B{\setminus}\{1\}}\), then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

On decomposability of finite groups

Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.     

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.     

Some properties of autocentral automorphisms of a group

Let G be a group, Aut(G) and L(G) denote the full automorphisms group and absolute centre of G, respectively. The automorphism \({\alpha\in Aut(G)}\) is called autocentral if \({g^{-1}\alpha(g)\in L(G)}\), for all \({g\in G}\). In the present paper, we investigate the properties of such automorphisms.     

Essential dimension of algebraic groups,including bad characteristic

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.     

Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups

Let G = N ? A, where N is a stratified group and A = ? acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian Δ on G. We prove a theorem of Mihlin–Hörmander type for spectral multipliers of Δ. The proof of the theorem hinges on a Calderón–Zygmund theory adapted to a sub-Riemannian structure of G and on L¹-estimates of the gradient of the heat kernel associated to the sub-Laplacian Δ.     

Nonstandard representations of locally compact groups

In the note, it is proved that, under natural conditions, any infinite-dimensional unitary representation T of a direct product of groups G = K × N, where K is a compact group and N is a locally compact Abelian group, is imaged by a representation of the nonstandard analog \(\tilde G\) of the group G in the group of nonstandard matrices of a fixed nonstandard size.     

On profinite groups with word values covered by nilpotent subgroups

Let N stand for the class of nilpotent groups or one of its well-known generalizations. For a multilinear commutator word w and a profinite group G we show that w(G) is finite-by-N if and only if the set of wvalues in G is covered by countably many finite-by-N subgroups. Earlier this was known only in the case where w = x or w = [x, y].     

Classification of finite groups satisfying a minimal condition

If H is a subgroup of a finite group G then we denote the normal closure of H in G by H ^G . We call G a PE-group if every minimal subgroup X of G satisfies N ^G (X) ∩ X ^G = X. The authors classify the finite non-PE-groups whose maximal subgroups of even order are PE-groups.     

Finite simple groups that are not spectrum critical

Let G be a finite group. The spectrum of G is the set ω(G) of orders of all its elements. The subset of prime elements of ω(G) is called the prime spectrum and is denoted by π(G). A group G is called spectrum critical (prime spectrum critical) if, for any subgroups K and L of G such that K is a normal subgroup of L, the equality ω(L/K) = ω(G) (π(L/K) = π(G), respectively) implies that L = G and K = 1. In the present paper, we describe all finite simple groups that are not spectrum critical. In addition, we show that a prime spectrum minimal group G is prime spectrum critical if and only if its Fitting subgroup F(G) is a Hall subgroup of G.     

On groups with relatively small normalizers of nonabelian subgroups

We investigate the structure of a finite nonsolvable group G in which the index |N _G (A): AC _G (A)| is either one or a prime for any nonabelian subgroup A.     

Varieties of Characters

Let G be a connected reductive affine algebraic group. In this short note we define the variety of G-characters of a finitely generated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group.     

Length-type parameters of finite groups with almost unipotent automorphisms

Let α be an automorphism of a finite group G. For a positive integer n, let E _G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E _G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E _G,n (α). For soluble G we prove that if, for some n, the Fitting height of E _G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F _h* (H) = H, where F ₀* (H) = 1, and F _i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F _i *(H)). Let m be the number of prime factors of the order |α| counting multiplicities. It is proved that if, for some n, the generalized Fitting height E _G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE _G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

Extremely primitive groups and linear spaces

A non-regular primitive permutation group is called extremely primitive if a point stabilizer acts primitively on each of its nontrivial orbits. Let S be a nontrivial finite regular linear space and G ≤ Aut(S). Suppose that G is extremely primitive on points and let rank(G) be the rank of G on points. We prove that rank(G) ≥ 4 with few exceptions. Moreover, we show that Soc(G) is neither a sporadic group nor an alternating group, and G = PSL(2, q) with q + 1 a Fermat prime if Soc(G) is a finite classical simple group.     

A note on weakly-supplemented subgroups of finite groups

A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the paper, we extend one main result of Kong and Liu (2014).     

On Stanley’s inequalities for character multiplicities

Let G be a group of automorphisms of a ranked poset \({{\mathcal Q}}\) and let N _k denote the number of orbits on the elements of rank k in \({{\mathcal Q}}\). What can be said about the N _k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.     

Finite groups with some non-Abelian subgroups of non-prime-power order

Let G be a finite group and NA(G) denote the number of conjugacy classes of all nonabelian subgroups of non-prime-power order of G. The Symbol π(G) denote the set of the prime divisors of |G|. In this paper we establish lower bounds on NA(G). In fact, we show that if G is a finite solvable group, then NA(G) = 0 or NA(G) ≥ 2^|π(G)|?2, and if G is non-solvable, then NA(G) ≥ |π(G)| + 1. Both lower bounds are best possible.