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Let F be a field of characteristic zero and let A be an F -algebra graded by a finite group G of order k. Given a non-negative integer n and a sum n = n 1 + ⋯ + n k of k non-negative integers, we associate a S 〈 n 〉 -module to A , where S 〈 n 〉 : = S n 1 × ⋯ × S n k , and we denote its S 〈 n 〉 -character by χ 〈 n 〉 (A). In this paper, for all sum n = n 1 + ⋯ + n k , we make explicit the decomposition of χ 〈 n 〉 (A) for some important G -graded algebras A and we compute the number l n G (A) of irreducibles appearing in all such decompositions. Our main goal is to classify G -graded algebras A such that the sequence l n G (A) is bounded by three. [ABSTRACT FROM AUTHOR]

For a p -group S , we define a new characteristic subgroup Q (e) (S) of S. We prove that Q (e) (S) is also a characteristic subgroup for p -stable finite groups of characteristic p having S as a Sylow subgroup. The present paper also establishes that Q (e) (S) is a normal subgroup of any p -stable fusion system over S. [ABSTRACT FROM AUTHOR]

Finite groups in which every element has prime power order (EPPO-groups) are nowadays fairly well understood. For instance, if G is a soluble EPPO-group, then the Fitting height of G is at most 3 and | π (G) | ⩽ 2 (Higman, 1957). Moreover, Suzuki showed that if G is insoluble, then the soluble radical of G is a 2-group and there are exactly eight nonabelian simple EPPO-groups. In the present work we concentrate on finite groups in which every commutator has prime power order (CPPO-groups). Roughly, we show that if G is a CPPO-group, then the structure of G ′ is similar to that of an EPPO-group. In particular, we show that the Fitting height of a soluble CPPO-group is at most 3 and | π (G ′) | ⩽ 3. Moreover, if G is insoluble, then R (G ′) is a 2-group and G ′ / R (G ′) is isomorphic to a simple EPPO-group. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes. A subgroup A of a finite group G is said to be: (i) σ-subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i − 1 ⊴ A i or A i / (A i − 1) A i is a σ j -group, j = j (i) , for all i = 1 , ... , n ; (ii) modular in G if the following conditions are held: (1) 〈 X , A ∩ Z 〉 = 〈 X , A 〉 ∩ Z for all X ≤ G , Z ≤ G such that X ≤ Z , and (2) 〈 A , Y ∩ Z 〉 = 〈 A , Y 〉 ∩ Z for all Y ≤ G , Z ≤ G such that A ≤ Z ; (iii) σ-quasinormal in G if A is σ -subnormal and modular in G. We obtain a description of finite groups in which σ -quasinormality (respectively, modularity) is a transitive relation. Some known results are extended. [ABSTRACT FROM AUTHOR]

For a finite dimensional representation V of a finite reflection group W , we consider the rational Cherednik algebra H t , c (V , W) associated with (V , W) at the parameters t ≠ 0 and c. The Dunkl total angular momentum algebra O t , c (V , W) arises as the centraliser algebra of the Lie superalgebra osp (1 | 2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of H t , c (V , W) and the Clifford algebra generated by V. We show that when dim ⁡ V ≥ 3 and for every value of the parameter c , the centre of O t , c (V , W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (− 1) V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into O t , c (V , W) , we establish results analogous to "Vogan's conjecture" for a family of operators depending on suitable elements of the double cover W ˜. [ABSTRACT FROM AUTHOR]

The directed power graph P → (G) of a group G is the simple digraph with vertex set G such that x → y if y is a power of x. The power graph of G , denoted by P (G) , is the underlying simple graph. The enhanced power graph P e (G) of G is the simple graph with vertex set G in which two elements are adjacent if they generate a cyclic subgroup. In this paper, it is proven that, if two groups have isomorphic power graphs, then they have isomorphic enhanced power graphs, too. It is known that any finite nilpotent group of order divisible by at most two primes has perfect enhanced power graph. We investigated whether the same holds for all finite groups, and we have obtained a negative answer to that question. Further, we proved that, for any n ≥ 0 and prime numbers p and q , every group of order p n q and p 2 q 2 has perfect enhanced power graph. We also give a complete characterization of symmetric and alternative groups with perfect enhanced graphs. Finally, we briefly paid attention to the difference graph, which is a difference between the enhanced graph and power graph and proved that groups whose difference graphs are perfect, have perfect enhanced graphs. [ABSTRACT FROM AUTHOR]

The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs K t , s with s > (t - 1) ! . In this note we deepen their understanding further by determining their automorphism group. [ABSTRACT FROM AUTHOR]

Let M=(M,\omega) be either S^2 \times S^2 or \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form \omega. Suppose a finite cyclic group \mathbb {Z}_n is acting effectively on (M,\omega) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism \mathbb {Z}_n\hookrightarrow Ham(M,\omega). In this paper, we investigate the homotopy type of the group Symp^{\mathbb {Z}_n}(M,\omega) of equivariant symplectomorphisms. We prove that for some infinite families of \mathbb {Z}_n actions satisfying certain inequalities involving the order n and the symplectic cohomology class [\omega ], the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczyński classification of smooth \mathbb {Z}_n-actions on Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

We consider the problem of whether, for a given virtually torsionfree discrete group \Gamma, there exists a cocompact proper topological \Gamma-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. This problem is related to Nielsen Realization. We will make the assumption that the expected manifold model has a zero-dimensional singular set. Then we solve the problem in the case, for instance, that \Gamma contains a normal torsionfree subgroup \pi such that \pi is hyperbolic and \pi is the fundamental group of an aspherical closed manifold of dimension greater or equal to five and \Gamma /\pi is a finite cyclic group of odd order. [ABSTRACT FROM AUTHOR]

The (co)homological dimension of a homomorphism \phi :G\to H is the maximal number k such that the induced homomorphism in k-th (co)homology groups is nonzero for coefficients in some H-module. It is known that for geometrically finite groups G, cd(G)=hd(G) and cd(G\times G)=2cd(G). We prove analogous theorems for homomorphisms of geometrically finite groups. The analogy stops working on the Eilenberg-Ganea equality cd(G)=gd(G) where cd(G)>2 and gd(G) is the geometric dimension of G. We show that for every k>2 there is a group homomorphism \phi _k:\pi _k\to \mathbb {Z}^k with cd(\phi _k)

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction. [ABSTRACT FROM AUTHOR]

A group G is called residually finite if for every non-trivial element g ∈ G , there exists a finite quotient Q of G such that the element g is non-trivial in the quotient as well. Instead of just investigating whether a group satisfies this property, a new perspective is to quantify residual finiteness by studying the minimal size of the finite quotient Q depending on the complexity of the element g , for example by using the word norm ‖ g ‖ G if the group G is assumed to be finitely generated. The residual finiteness growth RF G : N → N is then defined as the smallest function such that if ‖ g ‖ G ≤ r , there exists a morphism φ : G → Q to a finite group Q with | Q | ≤ RF G (r) and φ (g) ≠ e Q. Although upper bounds have been established for several classes of groups, exact asymptotics for the function RF G are only known for very few groups such as abelian groups, the Grigorchuk group and certain arithmetic groups. In this paper, we show that the residual finiteness growth of virtually abelian groups equals log k for some k ∈ N , where the value k is given by an explicit expression. As an application, we show that for every m ≥ 1 and every 1 ≤ k ≤ m , there exists a group G containing a normal abelian subgroup of rank m and with RF G ≈ log k. [ABSTRACT FROM AUTHOR]

Let G be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual p -finiteness of G for proving that this generalized free product is residually a finite nilpotent group or residually a finite metanilpotent group. This approach can be applied under most of the conditions on the amalgamated subgroup that allow the study of residual p -finiteness. Namely, we consider the cases where the amalgamated subgroup is a) periodic, b) locally cyclic, c) central in one of the free factors, d) normal in both free factors, or e) is a retract of one of the free factors. In each of these cases, we give certain necessary and sufficient conditions for G to be residually a) a finite nilpotent group, b) a finite metanilpotent group. [ABSTRACT FROM AUTHOR]

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra \mathbb {Q}G for G a finite generalized strongly monomial group. For the same groups with no exceptional simple components in \mathbb {Q}G, we describe a subgroup of finite index in the group of units \mathcal {U}(\mathbb {Z}G) of the integral group ring \mathbb {Z}G that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable. [ABSTRACT FROM AUTHOR]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2 n < 250 , specifically, the question remains open only 7 values of n. [ABSTRACT FROM AUTHOR]

Let G(q) be a twisted or untwisted finite exceptional group of Lie type over GF (q) , either adjoint or simply connected, and let W be the Weyl group of G(q). This paper compiles an extensive data set on the maximal tori of G(q). A systematic labeling is given, via the corresponding conjugacy classes of W, together with a complete list of the structures of the maximal tori in terms of their torsion coefficient decomposition. Additionally the associated admissible graphs and the structure or shape of centralizers of elements in W are given as well as compatible computer files. Also various apparent inconsistencies in the current literature on simply connected groups are discussed and resolved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate whether the Uniqueness Theorem introduced by Feit and Thompson holds in finite simple groups of Lie type of Lie rank one. We also introduce and examine the Strong Uniqueness Theorem. We define the concept of containment minimality and prove that a non-Abelian finite simple group is minimal if and only if it is containment minimal. Finally, we inquire when a minimal non-soluble group has a faithful doubly transitive representation. [ABSTRACT FROM AUTHOR]

It is very difficult to determine the full automorphism group of compact Riemann and Klein surfaces. The automorphism groups of hyperelliptic surfaces, and those of (compact, non-orientable, unbordered) Klein surfaces whose genus is less or equal to 7 are known, but a lot of hard work is left to do. To amplify the knowledge in this field, in this paper the full automorphism groups of the surfaces of genus 8 are calculated. [ABSTRACT FROM AUTHOR]

The weak commutativity group χ (G) is generated by two isomorphic groups G and G φ subject to the relations [ g , g φ ] = 1 for all g ∈ G . We establish a finiteness criterion for the subgroups of χ (G) in terms of the set T χ (G) = { [ g 1 , g 2 φ ] | g i ∈ G } . We also obtain sufficient conditions under which some specific (local) classes of groups are invariant under the operator χ. Moreover, we prove that if G is a locally finite group with exp (G) = n , then χ (G) is locally finite and has finite n-bounded exponent. Communicated by Mark Lewis [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group. We say that a subgroup H of G is a CSS-subgroup in G, if there exists a normal subgroup K of G such that G=HK and H∩K is SS-quasinormal in G. In this paper, we study the p-nilpotency and supersolvability of G under the assumption that every member of a small subset of the set of all maximal subgroups of the Sylow p-subgroups P of G with certain property is a CSS-subgroup in G. Using this criterion, we obtain new results which generalize and unify recent results in the literature. [ABSTRACT FROM AUTHOR]

The intersection non-simple graph, denoted by INS(G), of a finite abelian group G is an undirected graph whose vertex set is the collection of all proper non-trivial subgroups of G, and any two distinct vertices are adjacent if and only if their intersection is not a simple subgroup of G. We obtain some properties of INS(G) related to connectedness, completeness, degree, and girth. The concepts of bipartiteness, triangle-free, cluster, claw-free and cograph are taken into consideration. We also investigate the clique number, independence number, domination number, and planarity of INS(G). [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group, and g∈G. Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ(g)=0. Denote by Vo(G) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo(N)=Vo(G) is isomorphic to G. In this paper, we investigate the V-recognizability of E8(p), where p is a prime number. As an application, among the 610 primes p with p<10000 and p≡0,1,4 ( mod 5), we obtain that the method is always valid for confirming the V-recognizability of E8(p) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589. [ABSTRACT FROM AUTHOR]

Pontryagin duality provides a powerful tool for analyzing the structure and properties of locally compact abelian groups, in particular finite abelian groups. Now much can be done with non-abelian groups via the construction of character groups. In this paper, we provide a set of conditions a group must satisfy to be realized as a balanced character group for a metacyclic group. Furthermore, we provide a classification for the balanced character groups of a particular class of metacyclic groups, the dicyclic groups. [ABSTRACT FROM AUTHOR]

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent. The expected behavior of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broué's conjecture, and we raise a new question related to the change of central character. We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two-variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas. [ABSTRACT FROM AUTHOR]

We show that a mod- ℓ -representation of a p -adic group arising from the analogue of Yu's construction is supercuspidal if and only if it arises from a supercuspidal representation of a finite reductive group. This has been previously shown by Henniart and Vigneras under the assumption that the second adjointness holds. [ABSTRACT FROM AUTHOR]

We apply the Fixed Point Theorem for the actions of finite groups on Bruhat–Tits buildings and their products to establish two results concerning the groups of points of reductive algebraic groups over polynomial rings in one variable, assuming that the base field is of characteristic zero. First, we prove that for a reductive k -group G , every finite subgroup of G (k [ t ]) is conjugate to a subgroup of G (k). This, in particular, implies that if k is a finite extension of the p -adic field Q p , then the group G (k [ t ]) has finitely many conjugacy classes of finite subgroups, which is a well-known property for arithmetic groups. Second, we give a short proof of the theorem of Raghunathan–Ramanathan [26] about G -torsors over the affine line. [ABSTRACT FROM AUTHOR]

This paper presents the findings regarding the displacement-load behavior of both plane strain model experiments' ultimate load and numerical analyses conducted on unreinforced and reinforced sand slopes loaded with strip footing. The investigated parameters include the impact of the reinforcement and varying eccentricity on the ultimate load and displacement of the strip footing. A group of finite element analyses was performed with the 3D plane strain model and the computer code ANSYS software to validate the results of the model experiments on a slope. The results from both the numerical analyses and model experiments suggested that the use of reinforcement could enhance the load–displacement behavior of the central and eccentrically loaded footings. The load–displacement curves demonstrated that a higher load eccentricity leads to a reduction in the ultimate load of the strip footing. The concordance between the computed and observed results was reasonably satisfactory for the load displacement and the overall behavioral trend. [ABSTRACT FROM AUTHOR]

In this paper, we study the derivations of group algebras of some important groups, namely, Dihedral (D2n), Dicyclic (T4n) and Semi-dihedral (SD8n). First, we explicitly classify all inner derivations of a group algebra 픽G of a finite group G over an arbitrary field 픽. Then we classify all 픽-derivations of the group algebras 픽D2n, 픽T4n and 픽(SD8n) when 픽 is a field of characteristic 0 or an odd rational prime p by giving the dimension and an explicit basis of these derivation algebras. We explicitly describe all inner derivations of these group algebras over an arbitrary field. Finally, we classify all derivations of the above group algebras when 픽 is an algebraic extension of a prime field. [ABSTRACT FROM AUTHOR]

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups Sn, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G, with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for Sn is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group. [ABSTRACT FROM AUTHOR]

Background: Modified Anterior Cervical Discectomy and Fusion with specific resection ranges is an effective surgical method for the treatment of focal ossification of the posterior longitudinal ligament (OPLL). Herein, we compare and analyse the static stress area distribution by performing different cuts on an original ideal finite element model. Method: A total of 96 groups of finite element models of the C4–C6 cervical spine with different vertebral segmentation ranges (width: 1–12 mm, height: 1–8 mm) were established. The same pressure direction and size were applied to observe the size and distribution area of stress following various ranges of excision of the C5 vertebral body. Results: Different cutting areas had similar stress aggregation points. As the contact area decreased, the stress and the bearing above area increased. The correlation of stress area variation was highest between the 1–2 MPa and 6 MPa–Max regions (Rho = − 0.975). In the surface visualisation model fitting, the width and height were of different ratios in different stress regions. The model with the best fitting degree was the 1–2 MPa group, and the equation fitting (Rho = 0.966) was as follows: Area = 908.80 − 25.92 × Width + 2.71 × Height. Conclusion: Modified Anterior Cervical Discectomy and Fusion with different resection ranges exhibited different stress areas. In a specific resection range of the cervical spine (1–12 mm, 0–8 mm), area conversion occurred at a threshold of 4 MPa. Additionally, the stress was concentrated at the contact points between the vertebral body and the rigid fixator. [ABSTRACT FROM AUTHOR]

Let \mathrm {L}^2_a(\mathbb {D}) be the classical Bergman space and let M_h denote the operator of multiplication by a bounded holomorphic function h. Let B be a finite Blaschke product of order n. An open question proposed by R. G. Douglas is whether the operators M_B on \mathrm {L}^2_a(\mathbb {D}) similar to \oplus _1^n M_z on \oplus _1^n \mathrm {L}^2_a(\mathbb {D})? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernel. Since the operator M_z^* is in Cowen-Douglas class B_1(\mathbb {D}) in many cases, Douglas question can be reformulated for operators in B_1(\mathbb {D}), and the answer is affirmative for many operators in B_1(\mathbb {D}). A natural question occurs for operators in Cowen-Douglas class B_n(\mathbb {D}) (n>1). In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class B_2(\mathbb {D}), and give a negative answer. [ABSTRACT FROM AUTHOR]

This article is dedicated to the asymptotic geometry of wreath products F ≀ H : = (⨁ H F) ⋊ H where F is a finite group and H is a finitely generated group. Our first main result says that a coarse map from a finitely presented one-ended group to F ≀ H must land at bounded distance from a left coset of H . Our second main result, building on the later, is a very restrictive description of quasi-isometries between two lamplighter groups on finitely presented one-ended groups. Third, we obtain a complete classification of these groups up to quasi-isometry. More precisely, given two finite groups F 1 , F 2 and two finitely presented one-ended groups H 1 , H 2 , we show that F 1 ≀ H 1 and F 2 ≀ H 2 are quasi-isometric if and only if either (i) H 1 , H 2 are non-amenable quasi-isometric groups and | F 1 | , | F 2 | have the same prime divisors, or (ii) H 1 , H 2 are amenable, | F 1 | = k n 1 and | F 2 | = k n 2 for some k , n 1 , n 2 ≥ 1 , and there exists a quasi- (n 2 / n 1) -to-one quasi-isometry H 1 → H 2 . This can be seen as far reaching extension of a celebrated work of Eskin-Fisher-Whyte who treated the case of H = Z . Our approach is however fundamentally different, as it crucially exploits the assumption that H is one-ended. Our central tool is a new geometric interpretation of lamplighter groups involving natural families of quasi-median spaces. [ABSTRACT FROM AUTHOR]

We develop the idempotent theory for algebras over a class of semigroups called left regular bands of groups (LRBGs), which simultaneously generalize group algebras of finite groups and left regular band (LRB) algebras. Our techniques weave together the representation theory of finite groups and LRBs, opening the door for a systematic study of LRBGs in an analogous way to LRBs. We apply our results to construct complete systems of primitive orthogonal idempotents in the Mantaci–Reutenauer algebra MR n [ G ] associated to any finite group G. When G is abelian, we give closed form expressions for these idempotents, and when G is the cyclic group of order two, we prove that they recover idempotents introduced by Vazirani. [ABSTRACT FROM AUTHOR]

For any finite group G, the Terwilliger algebra T(G) of the group association scheme satisfies the following inclusions: T0(G)⊆T(G)⊆T˜(G), where T0(G) is a specific vector space and T˜(G) is the centralizer algebra of the permutation representation of G induced by the action of conjugation. The group G is said to be triply transitive if T0(G)=T˜(G). In this paper, we determine the dimensions of T0(G) and T˜(G) for G being Tn,k=〈a,b∣a2n=1,an=b2,bab−1=ak〉, Cn⋊Cp and Cp⋊Cn, and show that Tn,k,Cn⋊C2 and C3⋊C2n are triply transitive. Additionally, we give a complete characterization of the Wedderburn components of the Terwilliger algebras of Tn,k, Cn⋊Cp and Cp⋊Cn when they are triply transitive. [ABSTRACT FROM AUTHOR]

We develop a theory of quasi post-critically finite degenerations of Blaschke products. This gives us tools to study the boundaries of hyperbolic components of rational maps in higher-dimensional moduli spaces. We use it to obtain a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component Hd, i.e., the hyperbolic component in the space of monic and centered polynomials that contains zd. We also show that the closure Hd is not a topological manifold with boundary for d ≥ 4 by constructing self-bumps on its boundary. [ABSTRACT FROM AUTHOR]

For an irreducible character χ of a finite group G, the codegree of χ is defined as | G : ker (χ) | / χ (1) . In this paper, we characterize finite p-groups with three character codegrees. [ABSTRACT FROM AUTHOR]

Let p be a prime and F p be a finite field of p elements. Let F p G denote the group algebra of the finite p-group G over the field F p and V (F p G) denote the group of normalized units in F p G . Suppose that G is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p , n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V (F p G) p l and G p l , Ω l (V (F p G)) and Ω l (G) are given. Furthermore, there is a direct proof for V (F p G) p ∩ G = G p . [ABSTRACT FROM AUTHOR]

We compute the dimensions of Ext G n (V , W) for all irreducible modules V, W lying in r-blocks of cyclic defect whose Brauer tree is either a star or a line (open polygon), which in the line case extends a result of Dudas. We also make use of this information to aid in completing the same computation for all cross characteristic r-blocks of cyclic defect in rank one groups of Lie type. [ABSTRACT FROM AUTHOR]

The semidirect discrete logarithm problem (SDLP) is the following analogue of the standard discrete logarithm problem in the semidirect product semigroup G ⋊ End (G) for a finite semigroup G. Given g ∈ G , σ ∈ End (G) , and h = ∏ i = 0 t - 1 σ i (g) for some integer t, the SDLP (G , σ) , for g and h, asks to determine t. As Shor's algorithm crucially depends on commutativity, it is believed not to be applicable to the SDLP. For generic semigroups, the best known algorithm for the SDLP is based on Kuperberg's subexponential time quantum algorithm. Still, the problem plays a central role in the security of certain proposed cryptosystems in the family of semidirect product key exchange. This includes a recently proposed signature protocol called SPDH-Sign. In this paper, we show that the SDLP is even easier in some important special cases. Specifically, for a finite group G, we describe quantum algorithms for the SDLP in G ⋊ Aut (G) for the following two classes of instances: the first one is when G is solvable and the second is when G is a matrix group and a power of σ with a polynomially small exponent is an inner automorphism of G. We further extend the results to groups composed of factors from these classes. A consequence is that SPDH-Sign and similar cryptosystems whose security assumption is based on the presumed hardness of the SDLP in the cases described above are insecure against quantum attacks. The quantum ingredients we rely on are not new: these are Shor's factoring and discrete logarithm algorithms and well-known generalizations. [ABSTRACT FROM AUTHOR]

In this paper we focus our attention on matrix or operator-valued spherical functions associated to finite groups (G, K), where K is a subgroup of G. We introduce the notion of matrix-valued spherical functions on G associated to any K-type δ ∈ K ^ by means of solutions of certain associated integral equations. The main properties of spherical functions are established from their characterization as eigenfunctions of right convolution multiplication by functions in A [ G ] K , the algebra of K-central functions in the group algebra A[G]. The irreducible representations of A [ G ] K are closely related to the irreducible spherical functions on G. This allows us to study and compute spherical functions via the representations of this algebra. [ABSTRACT FROM AUTHOR]

Let p be a prime and G a pro-p group of finite rank that admits a faithful, self-similar action on the p-ary rooted tree. We prove that if the set {g ε G | gpn = 1} is a nontrivial subgroup for some n, then G is a finite p-group with exponent at most pn. This applies, in particular, to power abelian p-groups. [ABSTRACT FROM AUTHOR]

AbstractConsider a finite group G which acts on itself such that the relation ∼ on G, which is defined by x∼y iff x is a fixed point of y under this action, is both reflexive and symmetric. Let fix(G) represent the set of fixed points of G, i.e., fix(G)={x∈G | xg=x for all g∈G}. We associate with G a new graph using G∖fix(G) as its set of vertices, connecting vertices x,y∈G∖fix(G) iff x is a fixed point of y. This graph is referred to as the stabilizing graph of G and is denoted by Δs(G). Additionally, the complement of the stabilizing graph Δs(G), denoted by ∇s(G), is named the nonstabilizing graph of G. The aim of this study is to analyze various properties of the nonstabilizing graph ∇s(G) and to explore how this graph-theoretical concept relates to the broader understanding of group actions. [ABSTRACT FROM AUTHOR]

Let G be a finite group and ℒ(G) the subgroup lattice of G. A subgroup M of G is called: (i) modular in G, if M is a modular element (in the sense of Kurosh) of the lattice ℒ(G); (ii) submodular in G if G has a chain of subgroups M = M0 ≤ M0 ≤⋯ ≤ Mt = G, where Mi is modular in Mi+1 for all i = 0, 1,…,t − 1. If A is a subgroup of G, then we denote by AmG the subgroup of A, generated by all of its subgroups that are modular in G.We say that a subgroup A is N-modular in G (N ≤ G), if for some modular subgroup T of G, containing A, N avoids the pair (T,AmG), i.e. N ∩ T = N ∩ AmG.We prove that if G is a soluble finite group and each of its submodular subgroups is N-modular in G, where N is the nilpotent residual of G, then the lattice ℒ(G) is modular. [ABSTRACT FROM AUTHOR]

In this paper, we consider the group formed by finite renormalizations as an infinite-dimensional Lie group. It is demonstrated that for the finite renormalization of the gauge coupling constant, its generators L̂n with n≥1 satisfy the commutation relations of the Witt algebra and, therefore, form its subalgebra. The commutation relations are also written for the more general case when finite renormalizations are made for both the coupling constant and matter fields. We also construct the generator of the Abelian subgroup corresponding to the changes of the renormalization scale. The explicit expressions for the renormalization group generators are written in the case when they act on the β-function and the anomalous dimension. It is explained how the finite changes of these functions under the finite renormalizations can be obtained with the help of the exponential map. [ABSTRACT FROM AUTHOR]

We prove that any geometrically connected curve X$X$ over a field k$k$ is an algebraic K(π,1)$K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of Z/nZ$\mathbb {Z}/n\mathbb {Z}$‐modules, with n$n$ invertible in k$k$, is canonically isomorphic to the cohomology of its corresponding π1(X)$\pi _1(X)$‐module. To this end, we explicitly construct some Galois coverings of X$X$ corresponding to Galois coverings of the normalisation of its irreducible components. When k$k$ is finite or algebraically closed, we precisely describe finite quotients of π1(X)$\pi _1(X)$ that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products H1×H1→H2$\operatorname{H}^1\times \operatorname{H}^1\rightarrow \operatorname{H}^2$ and H1×H2→H3$\operatorname{H}^1\times \operatorname{H}^2\rightarrow \operatorname{H}^3$ in étale cohomology in terms of finite group cohomology. [ABSTRACT FROM AUTHOR]

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]