A remark on Gelfand–Graev characters for finite simple groups

The famous Gelfand–Graev character of a group of Lie type G is a multiplicity free character of shape ν ^G , where ν is a suitable degree 1 character of a Sylow p-subgroup and p is the defining characteristic of G. We show that, for an arbitrary non-abelian simple group G, if ν is a linear character of a Sylow p-subgroup of G such that ν ^G is multiplicity free, then G is isomorphic to either a group of Lie type in defining characteristic p, or to a group PSL(2, q), where either p = q + 1, or p = 2 and q + 1 or q ? 1 is a 2-power.     

On Huppert’s Conjecture for 3 D 4(q), q ≥ 3

Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type ³ D ₄(q), when q?≥?3.     

Tian’s invariant of the Grassmann manifold

We prove that Tian’s invariant on the complex Grassmann manifold G _p,q(?)is equal to 1/(p+ q).The method introduced here uses a Lie group of holomorphic isometries which operates transitively on the considered manifolds and a natural imbedding of (?¹ (?))^p in G _p,q (?).     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(²G₂(q)), where q = 3²ⁿ⁺¹ for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to ² G ₂(q). We infer that if G is a finite group satisfying |G| = |² G ₂(q)| and Γ(G) = Γ (² G ₂(q)) then G ? = ² G ₂(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.     

On nonabelian composition factors of a finite prime spectrum minimal group

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and G _x,y \(\underline \triangleleft \) G _x . P. Cameron raised the question about the validity of the equality G _x,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then G _x,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E ₆(q), ² E ₆(q), E ₇(q), and E ₈(q), then G _x,y = 1.     

The L p -L q analog of Morgan’s theorem on exponential solvable Lie groups

In this paper, we define an analog of the L ^p -L ^q Morgan’s uncertainty principle for any exponential solvable Lie group G (p, q ∈ [1,+∞]). When G is nilpotent and has a noncompact center, the proof of such an analog is given for p, q ∈ [2,+∞], extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular restrictive cases so far. We also prove the result for general exponential Lie groups with nontrivial center.     

Thompson’s conjecture for Lie type groups E 7(q)

In this article, we prove a conjecture of Thompson for an infinite class of simple groups of Lie type E ₇(q). More precisely, we show that every finite group G with the properties Z(G) = 1 and cs(G) = cs(E ₇(q)) is necessarily isomorphic to E ₇(q), where cs(G) and Z(G) are the set of lengths of conjugacy classes of G and the center of G respectively.     

Lie properties of symmetric elements in group rings II

Let F be a field of characteristic different from 2, and G a group with involution ∗. Write (FG)⁺ for the set of elements in the group ring FG that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and (FG)⁺ is Lie nilpotent (resp. Lie n-Engel), then FG is Lie nilpotent (resp. Lie m-Engel, for some m). Here, we classify the groups containing 2-elements such that (FG)⁺ is Lie nilpotent or Lie n-Engel.     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Characterization of G 2(q), where 2 < q ≡ −1(mod 3), by order components

We prove that the simple group G ₂(q), where 2 < q ≡ ?1(mod 3), is recognizable by the set of its order components. In other words, we prove that if G is a finite group with OC(G) = OC(G ₂(q)), then G ≌ G ₂(q).     

A note on transitive permutation groups of prime degree

LetG be a nonsolvable transitive permutation group of prime degreep. LetP be a Sylow-p-subgroup ofG and letq be a generator of the subgroup ofN _G(P) fixing one point. Assume that |N _G(P)|=p(p?1) and that there exists an elementj inG such thatj ^?1qj=q^(p+1)/2. We shall prove that a group that satisfies the above condition must be the symmetric group onp points, andp is of the form 4n+1.     

An L p -L q analog of miyachi’s theorem for nilpotent lie groups and sharpness problems

The purpose of this paper is to formulate and prove an L ^p -L ^q analog of Miyachi’s theorem for connected nilpotent Lie groups with noncompact center for 2 ≤ p, q ≤ +∞. This allows us to solve the sharpness problem in both Hardy’s and Cowling-Price’s uncertainty principles. When G is of compact center, we show that the aforementioned uncertainty principles fail to hold. Our results extend those of [1], where G is further assumed to be simply connected, p = 2, and q = +∞. When G is more generally exponential solvable, such a principle also holds provided that the center of G is not trivial. Representation theory and a localized Plancherel formula play an important role in the proofs.     

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n,q)

We denote by Gn the group of the upper unitriangular matrices over F_q, the finite field with q = p^t elements, and r(G_n) the number of conjugacy classes of G_n. In this paper, we obtain the value of r(G_n) modulo (q² -1)(q -1). We prove the following equalities     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

The bounded Lie Engel property on torsion group algebras

Let F be a field of characteristic different from 2 and G a group with involution ∗. Extend the involution to the group ring FG, and write (FG)⁻ for the Lie subalgebra of FG consisting of the skew elements. We classify the torsion groups G having no elements of order 2 such that (FG)⁻ is bounded Lie Engel.     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

On the cosets of the 2q-power group in the unit group modulop

Letp be a prime number ≡ 3 mod 4,G ^p the unit group of ?/p?, andg a generator ofG _p. Letq be an odd divisor ofp - 1 andG _p ^2q = {t ^2q;t ∈G _pthe subgroup of index2q inG _p. The groupG _p ² / _p ^2q consists of the classes \(\bar g^{2j} \) ,j = 0,...,q – 1. In this paper we study the ’excesses’ of the classes \(\bar g^{2j} \) in {l,...,(_p–l)/2}, i.e., the numbers \(\Phi _j = \left| {\left\{ {k;1 \leqslant k \leqslant \left( {p - 1} \right)/2,\bar k \in \bar g^{2j} } \right\}} \right| - \left| {\left\{ {k;\left( {p - 1} \right)/2 \leqslant k \leqslant p - 1,\bar k \in \bar g^{2j} } \right\}} \right|\) ,j = 0.....q — 1. First we express therelative class number h _2q of the subfieldK _2q? ?(e^2#x03C0;i/p ) of degree [K _2q: ?] =2q in terms of these excesses. We use this formula to establish certaincongruences for the Ф^j. E.g., ifq ∈ {3,5,11}, each number Фj is congruent modulo 4 to each other iff 2 dividesh _2q ^- . Finally we study thevariance of the excesses, i.e., the number \(\sigma ^2 = ((\Phi _0 - \hat \Phi )^2 + \ldots + (\Phi _{q - 1} - \hat \Phi )^2 )/(q - 1)\) , where \(\hat \Phi \) is the mean value of the numbers Ф_j. We obtain an explicit lower bound for σ² in terms ofh _2q ^- /h ₂ ^- . Moreover, we show that log σ² is asymptotically equal to 21og(h _2q ^- h ₂ ^- )/(q - 1) forp→∞. Three tables illustrate the results.     

Lie properties of symmetric elements in group rings

Let ¹ be an involution of a group G extended linearly to the group algebra KG. We prove that if G contains no 2-elements and K is a field of characteristic $p\ne 2$, then the 1-symmetric elements of KG are Lie nilpotent (Lie n-Engel) if and only if KG is Lie nilpotent (Lie n-Engel).     

Factorial representations of path groups

The reduction of the energy representation of the group of mappings from I = [0, 1], S¹1, $\text{R}$₊ or $\text{R}$ into a compact semisimple Lie group G is given. For G = SU(2), the factoriality of the representation, which is of type III in the case I=$\text{R}$, is proved.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.