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In this article, we study a certain Galois property of subextensions of k(Ators)$k(A_{\mathrm{tors}})$, the minimal field of definition of all torsion points of an abelian variety A$A$ defined over a number field k$k$. Concretely, we show that each subfield of k(Ators)$k(A_{\mathrm{tors}})$ that is Galois over k$k$ (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of k$k$. As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height. [ABSTRACT FROM AUTHOR]

Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex to a tree. For example, it is shown that the number of generators of the sandpile group is at most one less than the number of leaves in the tree. For trees on a fixed number of vertices, the paths and stars are shown to provide extreme behavior, not only for the number of generators, but also for the number of spanning trees, and for Tutte polynomial evaluations that count the recurrent sandpile configurations by their numbers of chips. [ABSTRACT FROM AUTHOR]

The given text is a technical article discussing the structure and properties of finitely generated Abelian n-ary groups. It explores the isomorphism and decomposition of these groups, as well as the invariants associated with them. The text presents theorems, corollaries, and proofs related to these concepts. It is a mathematical analysis that may require further study for a complete understanding. [Extracted from the article]

Let G be a finite group. We denote by ν1(G) the commuting probability that two randomly chosen elements of G are commuting. In this paper, we present a complete isoclinic classification of finite non-nilpotent group G with ν1(G) > 5 16. It is an improvement on a result of Lescot et al. in 2014. [ABSTRACT FROM AUTHOR]

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation x n = 1 x^{n}=1 in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer equations than unknowns. Instead of systems of equations, we consider arbitrary first-order formulae in the group language with constants. Our result substantially generalizes the Klyachko–Mkrtchyan theorem (2014) on this topic. [ABSTRACT FROM AUTHOR]

Let 퐺 be a group acting 2-transitively on the boundary of a locally finite tree, and exclude the situation (which is a genuine exception) where 퐺 has both P ⁢ Γ ⁢ L 3 ⁢ ( 4 ) \mathrm{P}\Gamma\mathrm{L}_{3}(4) and P ⁢ Γ ⁢ L 3 ⁢ ( 5 ) \mathrm{P}\Gamma\mathrm{L}_{3}(5) as local actions. We show that, for each half-tree T a T_{a} , the local action of the rigid stabilizer of T a T_{a} at the root of T a T_{a} contains the soluble residual of the point stabilizer of the local action of 퐺. In particular, 퐺 is locally prosoluble if and only if its local actions have soluble point stabilizers; if 퐺 is not locally prosoluble, then it has micro-supported action on the boundary. We also prove some strong restrictions on the local actions of end stabilizers in 퐺. These results are partly inspired by Radu’s classification of groups acting boundary-2-transitively on trees with local action containing the alternating group, and partly based on the author’s recent classification of finite permutation groups that preserve an equivalence relation, act faithfully on blocks and act transitively on pairs of points from different blocks. [ABSTRACT FROM AUTHOR]

We prove that, in a finite group, if every rational irreducible character has odd degree, then all rational elements are 2-elements, as it was originally conjectured by P. H. Tiep and H. P. Tong-Viet. [ABSTRACT FROM AUTHOR]

The power graph (G) of a finite group G is the simple graph with vertex set G , in which two distinct vertices are adjacent if one of them is a power of the other. For an integer n ≥ 2 , let C n denote the cyclic group of order n and let r be the number of distinct prime divisors of n. For r ≤ 3 , the minimum cut-sets of (C n) are characterized in [S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Vertex connectivity of the power graph of a finite cyclic group, Discrete Appl. Math.266 (2019) 259–271]. In this paper, for r ≥ 4 , we identify certain cut-sets of (C n) such that any minimum cut-set of (C n) must be one of them. [ABSTRACT FROM AUTHOR]

A proper subgroup H of a finite group G is called a second maximal subgroup of G if H is a maximal subgroup of every maximal subgroup M in G with H ≤ M. In this paper, we investigate the structure of the finite group G in which CG(x) is a second maximal subgroup for every non-central element x in G, and prove that either G is solvable or G/Z(G)≅PSL(2,rn), where r = 2 with n a prime or r = 3 with n an odd prime. [ABSTRACT FROM AUTHOR]

Schottky space Sg${\mathcal {S}}_{g}$, where g⩾2$g \geqslant 2$ is an integer, is a connected complex orbifold of dimension 3(g−1)$3(g-1)$; it provides a parametrization of the PSL2(C)${\rm PSL}_{2}({\mathbb {C}})$‐conjugacy classes of Schottky groups Γ$\Gamma$ of rank g$g$. The branch locus Bg⊂Sg${\mathcal {B}}_{g} \subset {\mathcal {S}}_{g}$, consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If [Γ]∈Bg$[\Gamma] \in {\mathcal {B}}_{g}$, then there is a Kleinian group K$K$ containing Γ$\Gamma$ as a normal subgroup of index some prime integer p⩾2$p \geqslant 2$. The structural description, in terms of Klein–Maskit Combination Theorems, of such a group K$K$ is completely determined by a triple (t,r,s)$(t,r,s)$, where t,r,s⩾0$t,r,s \geqslant 0$ are integers such that g=p(t+r+s−1)+1−r$g=p(t+r+s-1)+1-r$. For each such tuple (g,p;t,r,s)$(g,p;t,r,s)$, there is a corresponding cyclic‐Schottky stratum F(g,p;t,r,s)⊂Bg$F(g,p;t,r,s) \subset {\mathcal {B}}_{g}$. It is known that F(g,2;t,r,s)$F(g,2;t,r,s)$ is connected. In this paper, for p⩾3$p \geqslant 3$, we study the connectivity of these F(g,p;t,r,s)$F(g,p;t,r,s)$. [ABSTRACT FROM AUTHOR]

The non-commuting graph of a non-abelian group G with center Z(G) is a simple undirected graph whose vertex set is G∖Z(G) and two vertices x,y are adjacent if xy≠yx. In this paper, we compute Signless Laplacian spectrum and Signless Laplacian energy of non-commuting graphs of certain finite non-abelian groups. We obtain several conditions such that the non-commuting graph of G is Q-integral and observe relations between energy, Signless Laplacian energy and Laplacian energy. In addition, we look into the hyperenergetic and hypoenergetic properties of non-commuting graphs of finite groups. We also assess whether the same graphs are Q-hyperenergetic and L-hyperenergetic. [ABSTRACT FROM AUTHOR]

AbstractLet G and A be finite groups of relative coprime orders and A acts on G via automorphisms. In this paper, we investigate the p-nilpotency and solvability of finite groups in which every maximal A-invariant subgroup of a Sylow p-subgroup of G for some prime p is S-semipermutable. [ABSTRACT FROM AUTHOR]

We study the effect of hole doping on the Kitaev spin liquid (KSL) and find that for ferromagnetic (FM) Kitaev exchange K the system is very susceptible to the formation of a FM spin polarization. Through density matrix renormalization group simulations on finite systems, we uncover that the introduction of a single hole, corresponding to ≈1% hole doping for the system size we consider, with a hopping strength of just t ~ 0.28K is enough to disrupt fractionalization and polarize the spins in the [001] direction due to an order-by-disorder mechanism. Taking into account a material relevant FM anisotropic exchange Γ drives the polarization towards the [111] direction via a transition into a topological FM state with chiral magnon excitations. We develop a parton mean-field theory incorporating fermionic holons and bosonic magnons, which accounts for the doping induced FM phases and topological magnon excitations. We discuss experimental implications for Kitaev candidate materials. [ABSTRACT FROM AUTHOR]

We show that for many right‐angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non‐hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank ⩾3$\geqslant 3$, we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation. [ABSTRACT FROM AUTHOR]

Let V$V$ be a finite‐dimensional vector space equipped with a nondegenerate Hermitian form over a field K${\mathbb {K}}$. Let G(V)${\mathcal {G}}(V)$ be the graph with vertex set the one‐dimensional nondegenerate subspaces of V$V$ and adjacency relation given by orthogonality. We give a complete description of when G(V)${\mathcal {G}}(V)$ is connected in terms of the dimension of V$V$ and the size of the ground field K${\mathbb {K}}$. Furthermore, we prove that if dim(V)>4$\dim (V) > 4$, then the clique complex F(V)${\mathcal {F}}(V)$ of G(V)${\mathcal {G}}(V)$ is simply connected. For finite fields K${\mathbb {K}}$, we also compute the eigenvalues of the adjacency matrix of G(V)${\mathcal {G}}(V)$. Then, by Garland's method, we conclude that H∼m(F(V);k)=0$\tilde{H}_m({\mathcal {F}}(V);{\mathbb {k}}) = 0$ for all 0⩽m⩽dim(V)−3$0\leqslant m\leqslant \dim (V)-3$, where k${\mathbb {k}}$ is a field of characteristic 0, provided that dim(V)2⩽|K|$\dim (V)^2 \leqslant |{\mathbb {K}}|$. Under these assumptions, we deduce that the barycentric subdivision of F(V)${\mathcal {F}}(V)$ deformation retracts to the order complex of the certain rank selection of F(V)${\mathcal {F}}(V)$ that is Cohen–Macaulay over k${\mathbb {k}}$. Finally, we apply our results to the Quillen poset of elementary abelian p$p$‐subgroups of a finite group and to the study of geometric properties of the poset of nondegenerate subspaces of V$V$ and the poset of orthogonal decompositions of V$V$. [ABSTRACT FROM AUTHOR]

We consider word maps over finite groups. An n-variable word w is an element of the free group on n-symbols. For any group G, a word w induces a map from Gn → G where (g1,..., gn) →w(g1,..., gn). We observe that many groups have word maps that decompose into components. Such a decomposition facilitates a recursive approach to studying word maps. Building on this observation, and combining it with relevant properties of the word maps, allows us to develop an algorithm to calculate representatives of all the word maps over a finite group. Given these representatives, we can calculate word maps with specific properties over a given group, or show that such maps do not exist. In particular, we have computed an explicit a word on A5 such that only generating tuples are nontrivial in its image. We also discuss how our algorithm could be used to computationally address many open questions about word maps. Promising directions of potential applications include Amit's conjecture, questions of chirality and rationality, and the search for multilinear maps over a group. We conclude with open questions regarding these problems. [ABSTRACT FROM AUTHOR]

We give a short proof for an explicit upper bound on the proportion of permutations of a given prime order p, acting on a set of given size n, which is sharp for certain n and p. Namely, we prove that if n ≡ k (mod p) with 0 ≤ k ≤ p-1, then this proportion is at most (p ⋅ k!)-1 with equality if and only if p ≤ n < 2n. [ABSTRACT FROM AUTHOR]

Let G be a finite classical group generated by transvections, i.e., one of SL n (q) , SU n (q) , Sp 2 n (q) , or O 2 n ± (q) (q even) , and let X be a generating set for G containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the diameter of the Cayley graph Cay (G , X) is bounded by (n log ⁡ q) C for some constant C. This confirms Babai's conjecture on the diameter of finite simple groups in the case of generating sets containing a transvection. By combining this with a result of the author and Jezernik it follows that if G is one of SL n (q) , SU n (q) , Sp 2 n (q) and X contains three random generators then with high probability the diameter Cay (G , X) is bounded by n O (log ⁡ q). This confirms Babai's conjecture for non-orthogonal classical simple groups over small fields and three random generators. [ABSTRACT FROM AUTHOR]

Let 퐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G a parabolic subgroup and 푋 a homogeneous space of 퐺 admitting an equivariant projection on the flag variety G / Q with fibres given by copies of lattice quotients of a semisimple factor of 푄. Given a probability measure 휇, Zariski-dense in a copy of H = SL 2 ⁡ (R) in 퐺, we give a description of 휇-stationary probability measures on 푋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q , the type of stationary measures that 휇 admits depends strongly on the position of 퐻 relative to 푄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss. [ABSTRACT FROM AUTHOR]

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [ABSTRACT FROM AUTHOR]

A finite group G is said to be a B (n , k) group if for any n -element subset { a 1 , ... , a n } of G , | { a i a j   |   1 ≤ i , j ≤ n } | ≤ k. It is of interest to characterize the structure of B (n , k) groups for n (n + 1) / 2 ≤ k ≤ n 2 − 1. The B (5 , k) groups for 15 ≤ k ≤ 19 have been investigated by several authors. In this paper, we give a complete characterization of B (5 , 20) 2-groups by showing there are five classes of such groups which are nontrivial and nonabelian. [ABSTRACT FROM AUTHOR]

For a finite abelian group A , define f (A) to be the minimum integer such that for every complete digraph Γ on f vertices and every map w : E (Γ) → A , there exists a directed cycle C in Γ such that ∑ e ∈ E (C) w (e) = 0. The study of f (A) was initiated by Alon and Krivelevich (2021). In this article, we prove that f (Z p k) = O (p k (log ⁡ k) 2) , where p is prime, with an improved bound of O (k log ⁡ k) when p = 2. These bounds are tight up to a factor which is polylogarithmic in k. [ABSTRACT FROM AUTHOR]

In this article, we address the Amit–Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of [R. D. Camina, A. Iñiguez and A. Thillaisundaram, Word problems for finite nilpotent groups, Arch. Math. (Basel)115 (2020), 6, 599–609] by providing improved bounds for the case of finite nilpotent groups of class 2 for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of [A. Iñiguez and J. Sangroniz, Words and characters in finite 푝-groups, J. Algebra485 (2017), 230–246] using elementary techniques. [ABSTRACT FROM AUTHOR]

Let G be a finite group and Irr(G) be the set of irreducible characters of G. The number cod(χ) = | G : ker χ | / χ (1) is called the character codegree of χ ∈ Irr(G). In this paper, we show that if G has at most two composite character codegrees, then G is solvable. We also obtain sufficient numerical conditions on the character codegrees and character degrees for the solvability and supersolvability of a group. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be C S S -subgroup of G if there exists a normal subgroup K of G such that G = H K and H ∩ K is S S -quasinormal in G. In this paper, we investigate the p -nilpotency of the finite groups under the assumption that there is a subgroup D of P such that 1 < | D | < | P | and every subgroup of P with order | D | or 2 | D | (whenever | D | = 2) is C S S -subgroup, where p is a prime dividing the order of G and P is a Sylow p -subgroup of G. As an application of the results, a related problem posed by Heliel et al. in [Finite groups with minimal CSS-subgroups, Eur. J. Pure Appl. Math.14 (2021) 1002–1014] is solved. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a group G. Then H is called an IC Φ s − subgroup of G if (H ∩ [ H , G ]) H G / H G ⊆ Φ (H / H G) H sG / H G , where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We prove the p-supersolvability of a finite group G under the assumption that certain p-subgroups of G are IC Φ s − subgroups of G. Our main result generalizes and extends some recent work of Gao and Li (2022). [ABSTRACT FROM AUTHOR]

Let F be a saturated fusion system over a p-group S and let E be a subsystem of F with Aut E (S) = Aut F (S) . We prove that the fusion system F is generated by all E -morphisms and all F -automorphisms of some domestic intersections (called F ∖ E -domestic intersections in the paper) of F . We also get some results about control of fusion in finite groups as applications. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group. [ABSTRACT FROM AUTHOR]

Let G be a finite group, and let π be a set of primes. The aim of this paper is to obtain some results concerning how much information about the π -structure of G can be gathered from the knowledge of the sizes of conjugacy classes of its π -elements and of their multiplicities. Among other results, we prove that this multiset of class sizes determines whether G has a hypercentral Hall π -subgroup. [ABSTRACT FROM AUTHOR]

Let Γ be a graph with vertex set V, and let a, b be nonnegative integers. An (a, b)-regular set in Γ is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in V \ D has exactly b neighbours in D. In particular, a (1, 1)-regular set is called a total perfect code. Let G be a finite group and S a square-free subset of G closed under conjugation. The Cayley sum graph CayS (G , S) of G is the graph with vertex set G such that two vertices x, y are adjacent if and only if x y ∈ S . A subset (respectively, subgroup) D of G is called an (a, b)-regular set (respectively, subgroup (a, b)-regular set) of G if there exists a Cayley sum graph of G which admits D as an (a, b)-regular set. We obtain two necessary and sufficient conditions for a subgroup of a finite group G to be a total perfect code in a Cayley sum graph of G. We also obtain two necessary and sufficient conditions for a subgroup of a finite abelian group G to be a total perfect code of G. We classify finite abelian groups whose all non-trivial subgroups of even order are total perfect codes of the group, and as a corollary we obtain that a finite abelian group has the property that every non-trivial subgroup is a total perfect code if and only if it is isomorphic to an elementary abelian 2-group. We prove that, for a subgroup H of a finite abelian group G and any pair of positive integers (a, b) within certain ranges depending on H, H is an (a, b)-regular set of G if and only if it is a total perfect code of G. Finally, we give a classification of subgroup total perfect codes of a cyclic group, a dihedral group and a generalized quaternion group. [ABSTRACT FROM AUTHOR]

In the present paper, we study relative difference sets (RDSs) and linked systems of them. It is shown that a closed linked system of RDSs is always graded by a group. Based on this result, we also define a product of RDS linked systems sharing the same grading group. Further, we generalize the Davis-Polhill-Smith construction of a linked system of RDSs. Finally, we construct new linked system of RDSs in a Heisenberg group over a finite field and family of RDSs in an extraspecial p-group of exponent p 2 . All constructions of new RDSs and their linked systems make usage of cyclotomic Schur rings. [ABSTRACT FROM AUTHOR]

We compute all signatures of P S L 2 ( 7) and P S L 2 ( 1 1) which classify all orientation preserving actions of the groups P S L 2 ( 7) and P S L 2 ( 1 1) on compact, connected, orientable surfaces with orbifold genus ≥ 0. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. In this paper, for all odd numbers k≥3, using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

Let σ = { σ i ∣ i ∈ I } \sigma=\{\sigma_{i}\mid i\in I\} be some partition of the set of all primes and 퐺 a finite group. Then 퐺 is said to be 휎-full if 퐺 has a Hall σ i \sigma_{i} -subgroup for all i ∈ I i\in I and 휎-primary if 퐺 is a σ i \sigma_{i} -group for some 푖. In addition, 퐺 is 휎-soluble if every chief factor of 퐺 is 휎-primary and 휎-nilpotent if 퐺 is a direct product of 휎-primary groups. We write G N σ G^{\mathfrak{N}_{\sigma}} for the 휎-nilpotent residual of 퐺, which is the intersection of all normal subgroups 푁 of 퐺 with 휎-nilpotent G / N G/N . A subgroup 퐴 of 퐺 is said to be 휎-permutable in 퐺 provided 퐺 is 휎-full and 퐴 permutes with all Hall σ i \sigma_{i} -subgroups 퐻 of 퐺 (that is, A ⁢ H = H ⁢ A AH=HA ) for all 푖. And 퐴 is 휎-subnormal in 퐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G A=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G such that either A i − 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i − 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 휎-primary for all i = 1 , … , n i=1,\ldots,n . We prove that if 퐺 is a 휎-soluble group, then 휎-permutability is a transitive relation in 퐺 if and only if G N σ ∩ A G = G N σ ∩ A G G^{\mathfrak{N}_{\sigma}}\cap A^{G}=G^{\mathfrak{N}_{\sigma}}\cap A_{G} for every 휎-subnormal subgroup 퐴 of 퐺. [ABSTRACT FROM AUTHOR]

AbstractWe describe the full automorphism group of the directed reduced power graph and the undirected reduced power graph of a finite group. We compute the full automorphism groups of these graphs of several classes of finite groups. Also, we establish some relation between the automorphism group of the undirected reduced power graph (resp. the directed reduced power graph) and the automorphism group of the undirected power graph (resp. the directed power graph) of a finite group. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called an 풮풮ℋ-subgroup in G if there exists an s-permutable subgroup K of G such that HsG = HK and Hg ∩ N K(H) ≤ H for all g ∈ G, where HsG is the intersection of all s-permutable subgroup of G containing H. In this paper, we present two new sufficient and necessary conditions on the p-nilpotency of finite groups by use of a small quantity of 풮풮ℋ-subgroups of Sylow p-subgroups. A number of known results are improved and extended. [ABSTRACT FROM AUTHOR]

AbstractLet σ={σi|i∈I} be some partition of the set of all primes P, G a finite group and σ(G)={σi|σi∩π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi-subgroup of G for some i and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H-permutable if HA = AH for every member A∈H; m- H-permutable if H=〈A,B〉 for some modular subgroup A and H-permutable subgroup B of G; weakly m- H-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H∩T≤S≤H for some m-H-permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m-H-permutable in G. Some recent results are generalized and unified. [ABSTRACT FROM AUTHOR]

AbstractWe give a generic description of the non-defining primes l, for which the Steinberg character lies in the principal l-block of a finite reductive group. [ABSTRACT FROM AUTHOR]

A well-known conjecture of Gluck claims that | G : F ( G ) | ≤ b ( G ) 2 \lvert G:\mathbf{F}(G)\rvert\leq b(G)^{2} for all finite solvable groups 퐺, where F ⁢ ( G ) \mathbf{F}(G) is the Fitting subgroup and b ⁢ ( G ) b(G) is the largest degree of a complex irreducible character of 퐺. In this paper, we prove that Gluck’s conjecture holds for all wreath product type groups of the form G ≀ H 1 ≀ H 2 ≀ ⋯ ≀ H r G\wr H_{1}\wr H_{2}\wr\cdots\wr H_{r} , where 퐺 is a finite solvable group acting primitively on F ⁢ ( G ) / Φ ⁢ ( G ) \mathbf{F}(G)/\Phi(G) , and each H i H_{i} is a solvable primitive permutation group of finite degree. [ABSTRACT FROM AUTHOR]

We study the symbols of linear operators generated by a finite unitary group. Under some assumptions, it is shown that the operator can be restored from its symbol. We also introduce symbols of mixed unitary quantum channels generated by finite unitary groups. An example illustrating the technique is given. [ABSTRACT FROM AUTHOR]

By using the structure and some properties of extraspecial and generalized/almost extraspecial p -groups, we explicitly determine the number of elements of specific orders in such groups. As a consequence, one may find the number of cyclic subgroups of any (generalized/almost) extraspecial group. For a finite group G , the ratio of the number of cyclic subgroups to the number of subgroups is called the cyclicity degree of G and is denoted by cdeg (G) . We show that the set containing the cyclicity degrees of all finite groups is dense in [ 0 , 1 ] . This is equivalent to giving an affirmative answer to the following question posed by Tóth and Tărnăuceanu: "For every a ∈ [ 0 , 1 ] , does there exist a sequence (G n) n ≥ 1 of finite groups such that lim n → ∞ cdeg (G n) = a ?". We show that such sequences are formed of finite direct products of extraspecial groups of a specific type. [ABSTRACT FROM AUTHOR]