Subgroup Of GLn Research Articles

Galois groups of random additive polynomials

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.

Multi-groups

In the present paper we define homogeneous algebraic systems. Particular cases of these systems are semigroup (monoid, group) systems. These algebraic systems were studied by J. Loday, A. Zhuchok, T. Pirashvili, and N. Koreshkov. Quandle systems were introduced and studied by V. Bardakov, D. Fedoseev, and V. Turaev. We construct some group systems on the set of square matrices over a field k. Also, we define rack systems on the set V x G , where V is a vector space of dimension n over k and G is a subgroup of GLn(k). Finally, we find the connection between skew braces and dimonoids.

Degree bound for toric envelope of a linear algebraic group

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of degree at most the degree of the group as an algebraic variety. However, the degree of a linear algebraic group G ⊂ GL n ⁡ ( C ) G \subset \operatorname {GL}_n(C) can be arbitrarily large even for n = 1 n = 1 . One of the key ingredients of Hrushovski’s algorithm for computing the Galois group of a linear differential equation was an idea to “approximate” every algebraic subgroup of GL n ⁡ ( C ) \operatorname {GL}_n(C) by a “similar” group so that the degree of the latter is bounded uniformly in n n . Making this uniform bound computationally feasible is crucial for making the algorithm practical. In this paper, we derive a single-exponential degree bound for such an approximation (we call it a toric envelope), which is qualitatively optimal. As an application, we improve the quintuply exponential bound due to Feng for the first step of Hrushovski’s algorithm to a single-exponential bound. For the cases n = 2 , 3 n = 2, 3 often arising in practice, we further refine our general bound.

Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SL n (ℤ p )

Abstract Iwasawa algebras of compact p-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of p-adic Lie groups. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over ℤ p {\mathbb{Z}_{p}} which were uniform pro-p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime p > n + 1 {p>n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-p Iwahori subgroup of GL n ⁢ ( ℤ p ) {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro-p group.

On Higman`s Conjecture

Let Gn be the subgroup of GLn(q) consisting of the upper unitriangular matrices of size nxn over Fq. In 1960, G. Higman conjectured that the number of conjugacy classes of Gn, denoted by r(Gn), was given by a polynomial in q with integer coefficients. This has been verified for nn, r(Gn) can be expressed in terms of r(Gi), with i<n, and r(Tn), where Tn is the subset of primitive canonical matrices of Gn. Moreover, we determinate the expression of r(Tn) modulo (q-1)[(+1)/2]+3 and we prove that $r(Tn) mod(q-1)[(+1)/2]+3 is a polynomial in q with integer coefficients.

Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

Abstract Let A be a ring with 1 ≠ 0 ${1\neq 0}$ , not necessarily finite, endowed with an involution * ${*}$ , that is, an anti-automorphism of order ≤ 2 ${\leq 2}$ . Let H n ⁢ ( A ) ${H_{n}(A)}$ be the additive group of all n × n ${n\times n}$ hermitian matrices over A relative to * ${*}$ . Let 𝒰 n ⁢ ( A ) ${{\mathcal{U}}_{n}(A)}$ be the subgroup of GL n ⁢ ( A ) ${{\mathrm{GL}}_{n}(A)}$ of all upper triangular matrices with ones along the main diagonal. Let P = H n ⁢ ( A ) ⋊ 𝒰 n ⁢ ( A ) ${P=H_{n}(A)\rtimes{\mathcal{U}}_{n}(A)}$ , where 𝒰 n ⁢ ( A ) ${{\mathcal{U}}_{n}(A)}$ acts on H n ⁢ ( A ) ${H_{n}(A)}$ by * ${*}$ -congruence transformations. We may view P as a unipotent subgroup of either a symplectic group Sp 2 ⁢ n ⁢ ( A ) ${{\mathrm{Sp}}_{2n}(A)}$ , if * = 1 A ${*=1_{A}}$ (in which case A is commutative), or a unitary group U 2 ⁢ n ⁢ ( A ) ${{\mathrm{U}}_{2n}(A)}$ if * ≠ 1 A ${*\neq 1_{A}}$ . In this paper we construct and classify a family of irreducible representations of P over a field F that is essentially arbitrary. In particular, when A is finite and F = ℂ ${F={\mathbb{C}}}$ , we obtain irreducible representations of P of the highest possible degree.

Free subgroups in maximal subgroups of GLn(D)

ABSTRACTLet D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GLn(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of Mn(D) such that K*M, K∕F is Galois with Gal(K∕F)≅M∕K* and Gal(K∕F) is a finite simple group.

Expansive automorphisms of totally disconnected, locally compact groups

Abstract We study topological automorphisms α of a totally disconnected, locally compact group G which are expansive in the sense that ⋂ n ∈ ℤ α n ⁢ ( U ) = { 1 } $\bigcap_{n\in{\mathbb{Z}}}\alpha^{n}(U)=\{1\}$ for some identity neighbourhood U ⊆ G ${U\subseteq G}$ . Notably, we prove that the automorphism induced by an expansive automorphism α on a quotient group G / N ${G/N}$ modulo an α-stable closed normal subgroup N is always expansive. Further results involve the contraction groups U α := { g ∈ G : α n ⁢ ( g ) → 1 as n → ∞ } . $U_{\alpha}:=\{g\in G:\mbox{${\alpha^{n}(g)\to 1}$ as ${n\to\infty}$}\}.$ If α is expansive, then U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ is an open identity neighbourhood in G. We give examples where U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ fails to be a subgroup. However, U α ⁢ U α - 1 ${U_{\alpha}U_{\alpha^{-1}}}$ is an α-stable, nilpotent open subgroup of G if G is a closed subgroup of GL n ⁡ ( ℚ p ) ${\operatorname{GL}_{n}({\mathbb{Q}}_{p})}$ . Further results are devoted to the divisible and torsion parts of U α ${U_{\alpha}}$ , and to the so-called “nub” nub ⁡ ( α ) = U α ¯ ∩ U α - 1 ¯ ${\operatorname{nub}(\alpha)=\overline{U_{\alpha}}\cap\overline{U_{\alpha^{-1}}}}$ of an expansive automorphism.

Old Friends in Unexpected Places: Pascal (and Other) Matrices in GLn(ℂ)

One of the most beautiful results in early linear algebra is that every matrix over an algebraically closed field is similar to a Jordan matrix. This, of course, immediately proves that the Borel subgroup (the subgroup consisting of invertible n × n upper triangular matrices) is conjugate dense in GLn(ℂ). However, other matrices are also conjugate dense in GLn(ℂ). In this article, we show that a variation of Pascal matrices are one such class of conjugate dense matrices. We then use this fact to reduce finding matrix powers, roots, and inverses down to the corresponding problem for an appropriately chosen diagonal matrix. Having done this, we explore how one might create a wide array of other conjugate dense subgroups of GLn(ℂ).

Counting conjugacy classes for unitriangular matrices

Let Gn be the subgroup of GLn(q) consisting of upper unitriangular matrices. A longstanding conjecture, attributed to G. Higman, states that the number of conjugacy classes of Gn is given by a polynomial in q with integer coefficients. Recent results show that the resolution of this conjecture is linked to the classification of the canonical primitive connected matrices of Gn. In this paper, we calculate the number of canonical primitive connected matrices A of Gn satisfying that ΓA, the graph associated to A, has two pivot lines and there are, at most, two edges connecting the pivot lines. We prove that this number is a polynomial in q and it is connected with the combinatory of Catalan numbers.

A generalization of Catalan numbers

In this paper, we introduce a generalization of Catalan numbers. To obtain this extension, we construct a family of subsets which depend on three parameters and whose cardinals originate it. The elements of this family are used to classify canonical primitive connected matrices of the p-Sylow of GLn(q), problem that is related to Higman’s Conjecture, which asserts that if Gn is the subgroup of GLn(q) consisting of upper unitriangular matrices, then the number of conjugacy classes of Gn is a polynomial in q. The construction of these subsets allows us to prove by elementary way the recurrence relations and properties of our generalization of Catalan numbers. The associated sequences of integers can be arranged in tables called s-triangles. If s=1, the 1-triangle is the Catalan triangle. Consequently, to particularize the identities and properties of the s-triangles to the 1-triangle, we can deduce identities of Catalan numbers already proved. Moreover, for s≤5 the first diagonals of the s-triangles are well-known sequences of integers which arise in many mathematical scopes.

Normal coverings of linear groups

For a non-cyclic nite group G, let (G) denote the smallest number of conjugacy classes of proper subgroups of G needed to cover G. Let (G) denote the size of the largest set of conjugacy classes of G, such that any two elements from distinct classes generate G. In this paper several explicit bounds or formulas are given for (G) and (G), where G is a subgroup of GLn(q) containing SLn(q). The results hold also for the group G=Z(G). Motivated by questions in number theory, Bubboloni and Praeger have recently given bounds or exact formulas for (Sn) and (An), for all values of n. Further work of Bubboloni, Praeger and Spiga has established that (Sn) is bounded above and below by linear functions of n. This paper establishes a similar result for linear groups: it is shown that n= 2 2; the upper bound is exact in the case that n is an odd prime.

Cryptanalysis of a key exchange scheme based on block matrices

In this paper we describe a cryptanalysis of a key exchange scheme recently proposed by Álvarez, Tortosa, Vicent and Zamora. The scheme is based on exponentiation of block matrices over a finite field of prime order, and its security is claimed to rely in the hardness of a discrete logarithm problem in a subgroup of GLn(▪p). However, the proposal’s design allows for a clean attack strategy which exploits the fact that exponents are at some point added instead of multiplied as in a standard Diffie–Hellman construction. This strategy is moreover successful for a much more general choice of parameters than that put forward by Álvarez et al.

A Gelfand Model for Weyl Groups of Type D2n

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

Finite subgroups of algebraic groups

Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of GL n \operatorname {GL}_n over a field of any characteristic p p possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic p p , a commutative group of order prime to p p , and a p p -group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.

On the Number ofp-Regular Elements in Finite Simple Groups

AbstractAp-regular element in a finite group is an element of order not divisible by the prime numberp. We show that for every primepand every finite simple groupS, a fair proportion of elements ofSisp-regular. In particular, we show that the proportion ofp-regular elements in a finite classical simple group (not necessarily of characteristicp) is greater than 1/(2n), wheren– 1 is the dimension of the projective space on whichSacts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary fieldF, if the simple groupSis a quotient of a finite subgroup ofGLn(F) then for any primep, the proportion ofp-regular elements inSis at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups,p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for alln≥ 2 there exist infinitely many prime powersqsuch that the proportion of elements of odd order inPSL(n,q) is less than 3/√n.

Vector Invariants in Arbitrary Characteristic

Let k be an algebraically closed field of characteristic p ≥ 0. Let H be a subgroup of GLn(k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d vectors, d ≥ n, are obtained from those of n vectors by polarization. This result is not true when char k = p > 0 even in the case where H is a torus. However, we show that the algebra of invariants is always the p-root closure of the algebra of polarized invariants. We also give conditions for the two algebras to be equal, relating equality to good filtrations and saturated subgroups. As applications, we discuss the cases where H is finite or a classical group.

Subgroups of GLn(q) which fix sublattices of the subspace lattice

Subgroups of GLn(q) which fix sublattices of the subspace lattice

On (2, 3, 7)-generation of maximal parabolic subgroups

AbstractLet R be a ring with 1 and En (R) be the subgroup of GLn(R) generated by the matrices I + reij, r ∈ R, i ≠ j. We prove that the subgroup of consisting of the matrices of shape , where and , is (2, 3, 7)-generated whenever R is finitely generated and n, are large enough.

A Non-abelian Free Pro-pGroup Is Not Linear over a Local Field

In this paper we show that a (non-abelian) free pro-pgroup cannot be obtained as a closed subgroup ofGLn(F), whereFis a non-archimedean local field andnis arbitrary. Using a theorem of E. I. Zel'manov we deduce some group theoretic properties of linear pro-pgroups over a local field. Our main tool is a recent theorem by Pink characterizing compact subgroups ofGLn(F)