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Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of [Formula: see text]-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

In many fault detection problems, we want to identify all the d defective items from a set of n items using the minimum number of tests. Group testing is for the scenario where each test is on a subset of items, and tells whether the subset contains at least one defective item or not. In this paper, we study a variant of the classical group testing problem, which we call IGI (Identifying Good Items) problem. Instead of identifying all defective items, problem IGI is to identify good items that occupy no less than a prescribed proportion f 1 in the input set. We investigate group testing strategies for problem IGI using the minimum number of tests in the worst case. A class of testing strategies for problem IGI when d is unknown in advance is proposed, and a lower bound on the worst case number of tests required by any deterministic testing strategy for problem IGI is proved. The lower bound is valid no matter whether d is known in advance or not. When f approaches one, all the testing strategies in the proposed class are asymptotically optimal, and the best ratio between the upper bounds from the proposed class and the above lower bound is 3 log 2 3 .

We discuss necessary conditions for a compact quantum group to act on the algebra of noncommutative n -torus T θ n in a filtration preserving way in the sense of Banica and Skalski. As a result, we construct a family of compact quantum groups G θ = ( A θ n , Δ ) such that for each θ , G θ is the final object in the category of all compact quantum groups acting on T θ n in a filtration preserving way. We describe in detail the structure of the C*-algebra A θ n and provide a concrete example of its representation in bounded operators. Moreover, we show that A θ n is isomorphic to the C*-completed version of an algebra of multiple noncommutative torus. Finally, we compute the Haar measure of G θ and discuss its representation theory. For θ = 0 , the quantum group G 0 is nothing but the classical group T n ⋊ S n , where S n is the symmetric group.

How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the function lives on a finite non-commutative group G, and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of G. This can be applied to many functions that express interesting properties of G. To study these functions one wants to know how the different characters contribute to the sum? In this note we describe the $$G=SL_{2}({\mathbb {F}}_{q})$$ case of the theory we have been developing in recent years which attempts to give a fairly general answer to the above question for finite classical groups. The irreducible representations of $$SL_{2}({\mathbb {F}}_{q})$$ are “well known” for a very long time (Frobenius in Sitzber Preuss Akad Wiss 985–1021, 1896; Jordan in Am J Math 29:387–405, 1907; Schur in Journal fur die reine und angewandte Mathematik 132:85–137, 1907) and are a prototype example in many introductory courses on the subject. We are happy that we can say something new about them. In particular, it turns out that the representations that were considered as “anomalous” in the “old” point of view (known as the “philosophy of cusp forms”) are the building blocks of the current approach.

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

A group G is saturated with groups from a set ℜ of groups if every finite subgroup of G is contained in a subgroup of G that is isomorphic to some group in ℜ. Previously [Kourovka Notebook, Quest. 14.101], the question was posed whether a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in totality is itself a simple group of Lie type. A partial answer to this question is given for groups of Lie type of rank 1. We prove the following: Let a periodic group G be saturated with finite simple groups of Lie type of rank 1. Then G is isomorphic to a simple group of Lie type of rank 1 over a suitable locally finite field.

Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. We define a graph on π(G) with the following adjacency relation: different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(G). This graph is called the Gruenberg–Kegel graph or the prime graph of G and is denoted by GK(G). Let G and G 1 be two nonisomorphic finite simple groups of Lie type over fields of orders q and q 1, respectively, with different characteristics. It is proved that, if G is a classical group of a sufficiently high Lie rank, then the prime graphs of the groups G and G 1 may coincide only in one of three cases. It is also proved that, if G = A 1(q) and G 1 is a classical group, then the prime graphs of the groups G and G 1 coincide only if {G, G 1} is equal to {A 1(9), A 1(4)}, {A 1(9), A 1(5)}, {A 1(7), A 1(8)}, or {A 1(49),2 A 3(3)}.

We give a sufficient condition for complex manifolds for automorphism groups to become Lie groups. As an application, we see that the automorphism group of any strictly pseudoconvex domain or finite-type pseudoconvex domain has a Lie group structure.

A relationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and another finite classical group, namely the symplectic group Sp. New identities involving the number of partitions of n into parts of k different magnitudes are introduced in this context.

Let G be a Sylow p -subgroup of the unitary groups GU(3,q2)GU(3,q2), GU(4,q2)GU(4,q2), the symplectic group Sp(4,q)Sp(4,q) and, for q odd, the orthogonal group O+(4,q)O+(4,q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.

Since the ancient determination of the five platonic solids the study of symmetry and regularity has always been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many symmetries, i.e. large automorphism group G. In some important special situation higher degree of regularity means that G is an automorphism group of finite geometry. For example, a glance through the list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics and its applications such as coding theory, communication networks, and block design. An important tool for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that Aut GR = G depends very much on the description of the geometry with which one starts. For example, we can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3 special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form. The aim of this research is to develop an effective method for generation of graphs connected with classical geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The main approach is to avoid manual drawing and to calculate graph layout automatically according to its formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it becomes more complicated for graphs of geometrical nature. There are two main reasons for the investigations of spectra: (1) very often spectra carry much more useful information about the graph than a corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one could find some new Ramanujan graphs.

As a special type of factorization of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like $$MST_1$$MST1, $$MST_2$$MST2 and $$MST_3$$MST3. An LS with the shortest length, called a minimal logarithmic signature (MLS), is even desirable for cryptographic applications. The MLS conjecture states that every finite simple group has an MLS. Recently, the conjecture has been shown to be true for general linear groups $$GL_n(q)$$GLn(q), special linear groups $$SL_n(q)$$SLn(q), and symplectic groups $$Sp_n(q)$$Spn(q) with q a power of primes and for orthogonal groups $$O_n(q)$$On(q) with q a power of 2. In this paper, we present new constructions of minimal logarithmic signatures for the orthogonal group $$O_n(q)$$On(q) and $$SO_n(q)$$SOn(q) with q a power of an odd prime. Furthermore, we give constructions of MLSs for a type of classical groups--the projective commutator subgroup $$P{\varOmega }_n(q)$$PΩn(q).

Considering the automorphism groups of compact homogeneous spaces we inspect certain general properties, indicate a method for calculating the groups, and illustrate it with examples in a few particular cases.

An E-fuzzy group is a lattice-valued algebraic structure, defined on a crisp algebra which is not necessarily a group. The crisp equality is replaced by a particular fuzzy one - denoted by E. The classical group-like properties are formulated as appropriate fuzzy identities - special lattice-theoretic formulas. We prove basic features of E-fuzzy groups: properties of the unit and inverses, cancellability, solvability of equations, subgroup properties and others. We also prove that for every cut of an E-fuzzy group, which is a classical subalgebra of the underlying algebra, the quotient structure over the corresponding cut of the fuzzy equality is a classical group.

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p 2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p 2) of order 2|PSL(2,p 2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p 3. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.

Groups are the most fundamental and rich algebraic structure with respect to some binary operation in the study of algebra. In this paper, for the first time, we introduced the notion of neutrosophic triplet which is a group of three elements that satisfy certain properties with some binary operation. These neutrosophic triplets highly depends on the defined binary operation. Further, in this paper, we utilized these neutrosophic triplets to introduce the innovative notion of neutrosophic triplet group which is completely different from the classical group in the structural properties. A big advantage of neutrosophic triplet is that it gives a new group (neutrosophic triplet group) structure to those algebraic structures which are not group with respect to some binary operation in the classical group theory. In neutrosophic triplet group, we apply the fundamental law of Neutrosophy that for an idea A, we have neutral of A denoted as neut(a) and anti of A denoted as anti(A) to capture this beautiful picture of neutrosophic triplet group in algebraic structures. We also studied some interesting properties of this newly born structure. We further defined neutro-homomorphisms for neutrosophic triplet groups. A neutron-homomorphism is the generalization of the classical homomorphism with two extra conditions. As a further generalization, we gave rise to a new field or research called Neutrosophic Triplet Structures (such as neutrosophic triplet ring, neutrosophic triplet field, neutrosophic triplet vector space, etc.). In the end, we gave main distinctions and comparison of neutrosophic triplet group with the classical Molaei’s generalized group as well as the possible application areas of the neutrosophic triplet groups.

For the finite simple groups of twisted Lie types 2Al and 2Dl, we specify the description for the chief factors of a maximal parabolic subgroup which are involved in its unipotent radical. We prove a theorem in which, for every maximal parabolic subgroup of the groups 2Al(q2) and 2Dl(q2), we give the fragments of the chief series involving in the unipotent radical of this parabolic subgroup. The generators of the corresponding chief factors are presented in tables.

In group testing for complexes, the property of being positive is not defined on single items but on subsets of items. Given a set N of n items with at most d positive subsets of N, each positive subset having cardinality at most r, we need to determine the set P ? P ( N ) of positive subsets via group tests. In this paper, we give an adaptive algorithm for group testing for the complex model, for all r ? 2 , which can be seen as a generalization of the binary splitting method for classical group testing ( r = 1 ). For fixed r, our algorithm solves the problem with O ( d r log ? n ) tests while the best known non-adaptive algorithm requires O ( d r + 1 ( 1 + 3 / d ) d log ? n d ) tests. While non-adaptive solutions are preferable for some applications that require parallelization of tests, our adaptive algorithm is superior when d grows with n, yielding a polynomial number of tests on d and log ? k .

We prove that a periodic group, saturated with projective linear groups of dimension 2 over finite fields, is isomorphic to the projective linear group of dimension 2 over a locally finite field.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups., Comment: 19 pages

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

This paper deals with hypergroups, as a generalization of classical groups. An important tool in the theory of hyperstructures is the fundamental relation, which brings us into the classical algebra. In this paper we introduce the smallest equivalence relation ? ? on a given hypergroup H such that the quotient H / ? ? , the set of all equivalence classes, is an Engel group. We will characterize Engle groups via strongly regular relations and several results on the topic are presented.

Strongly real groups and totally orthogonal groups form two important subclasses of real groups. In this article we give a characterization of strongly real special 2-groups. This characterization is in terms of quadratic maps over fields of characteristic 2. We then provide examples of groups which are in one subclass and not the other., Comment: 16 pages

We provide a definition for an extended system of $\gamma$-factors for products of generic representations $\tau$ and $\pi$ of split classical groups or general linear groups over a non-archimedean local field of characteristic $p$. We prove that our $\gamma$-factors satisfy a list of axioms (under the assumption $p\neq 2$ when both groups are classical groups) and show their uniqueness (in general). This allows us to define extended local $L$-functions and root numbers. We then obtain automorphic $L$-functions $L(s,\tau \times \pi)$, where $\tau$ and $\pi$ are globally generic cuspidal automorphic representations of split classical groups or general linear groups over a global function field. In addition to rationality and the functional equation, we prove that our automorphic $L$-functions satisfy the Riemann Hypothesis.

The paper (Jiang, Automorphic forms: L-functions and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemporary mathematics, vol 614. American Mathematical Society, Providence, RI, 2014, pp 179–242) forms Part I of the theory of Automorphic Integral Transforms for Classical Groups, where the first named author made a conjecture on how the global Arthur parameters may govern the structure of the Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. This leads to a better understanding of the automorphic kernel functions with which the integral transforms yield conjecturally the endoscopic correspondences for classical groups. In this paper, we discuss the Twisted Automorphic Descent method and its variants that construct concrete modules for irreducible cuspidal automorphic representations of general classical groups. When the global Arthur parameters are generic, the details of the theory are referred to Jiang et al. (2015, accepted by IMRN), Jiang and Zhang (2015, submitted; 2015, in preparation), which extend the automorphic descent method of Ginzburg-Rallis-Soudry (The descent map from automorphic representations of GL(n) to classical groups. World Scientific, Singapore, 2011) to great generality.

In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree.

We consider the subgroups H in a linear or a unitary group G over a finite field such that Or(H) ≰ Z(G) for some odd prime r. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case.

Jeffrey D. Achter establishes in [2] a connection between the distribution of class groups of function fields and the distribution of eigenspaces in symplectic groups. Gunter Malle relates this idea in [10] to the number field case. Motivated by these texts we compute here for a finite p -torsion R-module H, where R is a Dedekind ring with finite quotients and q = | R / p | , the limit P G , ∞ , q , f ( H ) : = lim n → ∞ P G , n , q , f ( H ) : = lim n → ∞ | { g ∈ G n ( R / p f ) | ker ( g − 1 ) ≅ H } | | G n ( R / p f ) | for certain classical groups G of increasing dimension n. In doing so we extend the results of Jason Fulman ( [4] and [5] ) concerning distributions of eigenspaces over finite fields. Furthermore we give a reasonable backup for the conjecture of G. Malle (2.1 in [10] ).

In this paper, all the binary codes from the primitive permutation group P S U 2 ( 16 ) are constructed. It is shown that the groups P S U 2 ( 16 ) : 4 and S 17 are the automorphism groups of the constructed codes.

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra G 2 in characteristic 3 and in the dual of exceptional Lie algebra F 4 in characteristic 2 are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type B , C , F 4 in characteristic 2 and in the dual of Lie algebra of type G 2 in characteristic 3. In each case we give an explicit description of the nilpotent pieces in the dual defined in [2] , which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case-by-case definition in [11] , [26] in the case of classical groups and have nice properties independent of the characteristic of the base field. This completes the classification of nilpotent coadjoint orbits in the dual of Lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic.

Let (V, h) be a Hermitian space over a division algebra D which is of index at most two over a non-Archimedean local field k of residue characteristic not 2. Let G be the unitary group defined by h and let \({\sigma}\) be the adjoint involution. Suppose we are given two \({\sigma}\)-invariant but not \({\sigma}\)-fixed field extensions E1 and E2 of k in EndD(V) which are isomorphic under conjugation by an element g of G and suppose that there is a point x in the Bruhat–Tits building of G which is fixed by \({E_1^{\times}}\) and \({E_2^{\times}}\) in the reduced building of AutD(V). Then E1 is conjugate to E2 under an element of the stabilizer of x in G if E1 and E2 are conjugate under an element of the stabilizer of x in AutD(V) and a weak extra condition holds. In addition, in many cases the conjugation by g from E1 to E2 can be realized as conjugation by an element of the stabilizer of x in G. Further we give a concrete description of the canonical isomorphism from the set of \({E_1^\times}\) fixed points of the building of G onto the building of the centralizer of E1 in G.

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $\mu:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, $\mu^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of $\mu^{-1}(0)/\!/G$., Comment: 21 pages, final version, to appear in Math. Z

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by b(G). In this paper we calculate the base size of every primitive almost simple classical group with point stabilizer in Aschbacher’s collection S of irreducibly embedded almost simple subgroups. In this situation we also establish strong asymptotic results on the probability that randomly chosen subsets of Ω form a base for G. Indeed, with some specific exceptions, we show that almost all pairs of points in Ω are bases.

Let G = 〈 X 〉 ⩽ GL ( d , F ) be a classical group in its natural representation defined over a finite field F of even characteristic. We present Las Vegas algorithms to construct standard generators for G which permit us to write an element of G as a straight-line program in X, and to construct an involution as a straight-line program in X. If | F | > 4 , then the algorithms run in time polynomial in the size of the input, subject to the existence of a discrete logarithm oracle for F .

Let p 1, p 2, p 3 be primes. This is the second article in a series of three on the (p 1, p 2, p 3)-generation of the finite projective special unitary and linear groups PSU3(p n ), PSL3(p n ), where we say a noncyclic group is (p 1, p 2, p 3)-generated if it is a homomorphic image of the triangle group T p 1, p 2, p 3 . This paper is concerned with the case where p 1 = 2 and p 2 = p 3. We determine for any prime p 2 the prime powers p n such that PSU3(p n ) (respectively, PSL3(p n )) is a quotient of T = T 2, p 2, p 2 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU3(p n )) (respectively, Hom(T, PSL3(p n ))) is surjective as p n tends to infinity.

If m and n are non-negative integers, then three new classes of abelian p-groups are defined and studied: the m,n-simply presented groups, the m,n-balanced projective groups and the m,n-totally projec- tive groups. These notions combine and generalize both the theories of simply presented groups and p !+n -projective groups. If m,n = 0, these all agree with the class of totally projective groups, but when m+n � 1, they also include the p !+m+n -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) n-balanced projective groups considered in (15) and the n-summable groups considered in (2). The groups in these classes whose lengths are less than ! 2 are character- ized, and if in addition we have n = 0, they are determined by isometries of their p m -socles.

Langlands’ beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of \(\mathrm {GL}_n\) that stabilize a line in a given repesentation of \(\mathrm {GL}_n\). Such subgroups are said to be detected by the representation. In this paper we continue our study of the important special case where the representation of \(\mathrm {GL}_n\) is the triple tensor product representation \(\otimes ^3\). We prove a family of results describing when subgroups isomorphic to classical groups of type \(B_{n}\), \(C_n\), \(D_{2n}\) are detected.

Group testing is a well known search problem that consists in detecting the defective members of a set of objects O by performing tests on properly chosen subsets (pools) of the given set O. In classical group testing the goal is to find all defectives by using as few tests as possible. We consider a variant of classical group testing in which one is concerned not only with minimizing the total number of tests but aims also at reducing the number of tests involving defective elements. The rationale behind this search model is that in many practical applications the devices used for the tests are subject to deterioration due to exposure to or interaction with the defective elements. In this paper we consider adaptive, non-adaptive and two-stage group testing. For all three considered scenarios, we derive upper and lower bounds on the number of "yes" responses that must be admitted by any strategy performing at most a certain number t of tests. In particular, for the adaptive case we provide an algorithm that uses a number of "yes" responses that exceeds the given lower bound by a small constant. Interestingly, this bound can be asymptotically attained also by our two-stage algorithm, which is a phenomenon analogous to the one occurring in classical group testing. For the non-adaptive scenario we give almost matching upper and lower bounds on the number of "yes" responses. In particular, we give two constructions both achieving the same asymptotic bound. An interesting feature of one of these constructions is that it is an explicit construction. The bounds for the non-adaptive and the two-stage cases follow from the bounds on the optimal sizes of new variants of d-cover free families and (p, d)-cover free families introduced in this paper, which we believe may be of interest also in other contexts.

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. Furthermore, we describe how these results can be used to analyze part of Yal\c{c}inkaya's Black Box recognition algorithm for finite classical groups in odd characteristic., Comment: 28 pages

Let L2(q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L2(q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi.

In 2005, Ginzburg, Rallis and Soudry constructed, in terms of residues of certain Eisenstein series, and by use of the descent method, families of nontempered automorphic representations of \(Sp_{4nm} (\mathbb{A})\) and \(\widetilde{Sp}_{2n(2m - 1)} (\mathbb{A})\), which generalized the classical work of Piatetski-Shapiro on Saito-Kurokawa liftings. In this paper, we introduce a new framework (Diagrams of Constructions) in order to establish explicit relations among the representations introduced in [GRS05]. In particular, we prove that these constructions yield bijections between a certain set of cuspidal automorphic forms on \(\widetilde{Sp}_{2n} (\mathbb{A})\) and a certain set of square-integrable automorphic forms of \(Sp_{4n} (\mathbb{A})\). The proofs use new interpretations of composition of two consecutive descents with explicit identities, which we expect to be very useful to further investigation of the automorphic discrete spectrum of classical groups.

Given a group G which is not necessarily finite, and a Noetherian commutative ring R , an important question is when the cohomology ring H ∗ ( G , R ) is a Noetherian ring. For finite groups, this is the Venkov–Evens theorem. This theorem is extended to finite dimensional restricted Lie algebras when R is the ground field and to more general finite groups schemes over a field R by Friedlander and Suslin. Recently, van der Kallen and Touze have shown this for reductive algebraic groups over a field of positive characteristics with rational group cohomology. In this note, we prove a finite generation theorem for a large class of infinite groups which include all arithmetic subgroups of linear algebraic groups, lattices of semisimple Lie groups, mapping class groups of surfaces, outer automorphism groups of free groups, Gromov hyperbolic groups, outer automorphism groups of free groups, and many other natural infinite groups.

Let ( R , Λ ) be a commutative form ring, and let ( J , Γ ) be a form ideal of ( R , Λ ) . We obtain a complete description of all subgroups of the unitary groups U 2 n ( R , Λ ) which are normalized by relative elementary subgroup EU 2 n ( J , Γ ) for all n ≥ 4 .

Let G be one of the groups SL_n C, Sp_2n C, SO_m C, O_m C, or G_2. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G x P^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G one of the classical groups., Comment: 63 pages

Let G be a finite classical group defined over a finite field with odd characteristic. Let r > 2 be a prime, not dividing the characteristic, and D ⩽ G a Sylow r-subgroup. We consider the Frobenius category F D ( G ) and determine the cardinality of a minimal conjugation family for it. This amounts to determining the number of G-conjugacy classes of essential subgroups of D, that is, the essential rank of F D ( G ) . In addition, we characterise those classical groups which allow an r-local subgroup controlling r-fusion.

The maximal subgroups of the finite classical groups are divided into nine classes by Aschbacher's theorem. In this paper, we give explicit transcendental bases of the invariant subfields of those maximal subgroups of classical groups of linear, symplectic and unitary cases that lie in the first two of these classes. Also, we show that the invariant rings of the maximal subgroups of the finite linear groups that lie in the first class are polynomial rings.

Let I ( n ) be the number of involutions in a special orthogonal group SO ( n , F q ) defined over a finite field with q elements, where q is the power of an odd prime. Then the numbers I ( n ) form a semi-recursion, in that for m > 1 we have I ( 2 m + 3 ) = ( q 2 m + 2 + 1 ) I ( 2 m + 1 ) + q 2 m ( q 2 m − 1 ) I ( 2 m − 2 ) . We give a purely combinatorial proof of this result, and we apply it to give a universal bound for the character degree sum for finite classical groups defined over F q .

Let G be a classical group and suppose that G does not contains non-trivial graph automorphisms. In this paper we prove that the order complex of the coset poset of G is non-contractible. In order to prove it, we show that PG(−1) does not vanish, where PG(s) is the Dirichlet polynomial associated to the group G.

Let H be a simple finite classical group not isomorphic to PSL(n, q) for any n, q. We prove that every cyclic subgroup of H has a regular orbit on every nontrivial permutation H-set.