On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

Semisimple weakly symmetric pseudo-Riemannian manifolds

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature \((n-1,1)\) and trans-Lorentzian signature \((n-2,2)\).     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite 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.     

On R-conjugate-permutability of sylow subgroups

A subgroup H of a finite group G is said to be conjugate-permutable if HH^g = H^gH for all g ∈ G. More generaly, if we limit the element g to a subgroup R of G, then we say that the subgroup H is R-conjugate-permutable. By means of the R-conjugatepermutable subgroups, we investigate the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B under the condition that G = AB, where A and B are subgroups of G. Some results known in the literature are improved and generalized in the paper.     

On the <Emphasis Type="Italic">p</Emphasis>-supersolubility of a finite group with a µ-supplemented sylow <Emphasis Type="Italic">p</Emphasis>-subgroup

A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H ₁ K is a proper subgroup in G for every maximal subgroup H ₁ in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup.     

Positive-fraction intersection results and variations of weak epsilon-nets

A connected Finsler space (M, F) is said to be homogeneous if it admits a transitive connected Lie group G of isometries. A geodesic in a homogeneous Finsler space (G / H, F) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. In this paper, we study the problem of the existence of homogeneous geodesics on a homogeneous Finsler space, and prove that any homogeneous Finsler space of odd dimension admits at least one homogeneous geodesic through each point.     

Orlicz Algebras on Homogeneous Spaces of Compact Groups and Their Abstract Linear Representations

Let G be a compact group, H a closed subgroup of G and let m be the normalized G-invariant measure on the homogeneous space G / H obtained from Weil’s formula. In this article, for a given Young function \(\varphi \), we give a new class of Banach convolution algebras on homogeneous spaces of compact groups by introducing a convolution and an involution on the Orlicz space \(L^\varphi (G/H, m)\). Finally, a class of linear representations of this class of Banach convolution algebras is presented.     

On <Emphasis Type="Italic">ss</Emphasis>-quasinormal or weakly <Emphasis Type="Italic">s</Emphasis>-permutably embedded subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be an ss-quasinormal subgroup of G if there is a subgroup B of G such that \(G=HB\) and H permutes with every Sylow subgroup of B; H is said to be weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup \(H_{se}\) of G contained in H such that \(G=HT\) and \(H\cap T\le H_{se}\). We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is either ss-quasinormal or weakly s-permutably embedded in G. Some recent results are generalized and unified.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

<Emphasis Type="Italic">θ</Emphasis>-Pairs for 2-maximal subgroups of finite groups

Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) \({\langle H, A\rangle=G}\) and B = (A ∩ H) _G ; (ii) if A ₁/B is a proper subgroup of A/B and \({{A_1/B \vartriangleleft G/B}}\), then \({G\neq \langle H, A_1\rangle}\). In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

Joint spectrum and the infinite dihedral group

A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and H ^g are conjugate in the subgroup <H,H ^g >. We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = N _U (H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp_6n (q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.     

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).     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.     

The structure of finite groups with weakly <Emphasis Type="Italic">c</Emphasis>-normal subgroups

For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that \(G=HK\) and \(H\cap K\) is s-quasinormally embedded in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is weakly c-normal in G. Some recent results are generalized and unified.     

On weakly <Emphasis Type="Italic">S</Emphasis>Φ-supplemented subgroups of finite groups

Let G be a finite group. We say that a subgroup H of G is weakly SΦ-supplemented in G if G has a subgroup T such that G = HT and H∩T ≤ Φ(H)H_sG, where H_sG is the subgroup of H generated by all those subgroups of H that are s-permutable in G. In this paper, we investigate the influence of weakly SΦ-supplemented subgroups on the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.     

On coset-spaces of compact Lie groups by subgroups of corank 2

Let G be a simple compact connected simply connected Lie group, H its connected Lie subgroup of corank 2 which coincides with the commutator group of the centralizer of a torus, and let Sam(G/H) = 0. We prove that if a compact connected simply connected Lie group G' acts transitively and locally effectively on the manifold G/H, then G' is isomorphic to G. if the root system of G consists of roots of the same length, then the action of G' on G/H is similar to the action of G.