Elements ofG Research Articles

Asymptotics of the powers in finite reductive groups

AbstractLet 𝐺 be a connected reductive group defined overFq\mathbb{F}_{q}. Fix an integerM≥2M\geq 2, and consider the power mapx↦xMx\mapsto x^{M}on 𝐺. We denote the image ofG⁢(Fq)G(\mathbb{F}_{q})under this map byG⁢(Fq)MG(\mathbb{F}_{q})^{M}and estimate what proportion of regular semisimple, semisimple and regular elements ofG⁢(Fq)G(\mathbb{F}_{q})it contains. We prove that, asq→∞q\to\infty, the set of limits for each of these proportions is the same and provide a formula. This generalizes the well-known results forM=1M=1where all the limits take the same value 1. We also compute this more explicitly for the groupsGL⁢(n,q)\mathrm{GL}(n,q)andU⁢(n,q)\mathrm{U}(n,q)and show that the set of limits are the same for these two group, in fact, in bijection underq↦-qq\mapsto-qfor a fixed 𝑀.

Dense images of the power maps for a disconnected real algebraic group

Abstract Let 𝐺 be a complex algebraic group defined over ℝ, which is not necessarily Zariski-connected. In this article, we study the density of the images of the power maps g → g k g\to g^{k} , k ∈ N k\in\mathbb{N} , on real points of 𝐺, i.e., G ⁢ ( R ) G(\mathbb{R}) equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of G ⁢ ( R ) G(\mathbb{R}) . We also characterize surjectivity of the power map for a disconnected group G ⁢ ( R ) G(\mathbb{R}) . The results are applied in particular to describe the image of the exponential map of G ⁢ ( R ) G(\mathbb{R}) .

Interval Garside structures for the complex braid groups 𝐵(𝑒,𝑒,𝑛)

We define geodesic normal forms for the general series of complex reflection groups G ( e , e , n ) G(e,e,n) . This requires the elaboration of a combinatorial technique in order to explicitly determine minimal word representatives of the elements of G ( e , e , n ) G(e,e,n) over the generating set of the presentation of Corran–Picantin. Using these geodesic normal forms, we construct intervals in G ( e , e , n ) G(e,e,n) that are proven to be lattices. This gives rise to interval Garside groups. We determine which of these groups are isomorphic to the complex braid group B ( e , e , n ) B(e,e,n) and get a complete classification. For the other Garside groups that appear in our construction, we provide some of their properties and compute their second integral homology groups in order to understand these new structures.

On the characters of unipotent representations of a semisimple 𝑝-adic group

Let G G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K K and let V V be a unipotent representation of G ( K ) G(K) (for example, an Iwahori-spherical representation). We calculate the character of V V at compact very regular elements of G ( K ) G(K) .

Jordan-Chevalley decomposition in finite dimensional Lie algebras

Let g \mathfrak {g} be a finite dimensional Lie algebra over a field k k of characteristic zero. An element x x of g \mathfrak {g} is said to have an abstract Jordan-Chevalley decomposition if there exist unique s , n ∈ g s,n\in \mathfrak {g} such that x = s + n x=s+n , [ s , n ] = 0 [s,n]=0 and given any finite dimensional representation π : g → g l ( V ) \pi :\mathfrak {g}\to \mathfrak {gl}(V) the Jordan-Chevalley decomposition of π ( x ) \pi (x) in g l ( V ) \mathfrak {gl}(V) is π ( x ) = π ( s ) + π ( n ) \pi (x)=\pi (s)+\pi (n) . In this paper we prove that x ∈ g x\in \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if x ∈ [ g , g ] x\in [\mathfrak {g},\mathfrak {g}] , in which case its semisimple and nilpotent parts are also in [ g , g ] [\mathfrak {g},\mathfrak {g}] and are explicitly determined. We derive two immediate consequences: (1) every element of g \mathfrak {g} has an abstract Jordan-Chevalley decomposition if and only if g \mathfrak {g} is perfect; (2) if g \mathfrak {g} is a Lie subalgebra of g l ( n , k ) \mathfrak {gl}(n,k) , then [ g , g ] [\mathfrak {g},\mathfrak {g}] contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie’s Theorem, in addition to the fundamental theorems of Ado and Levi.

Non-abelian sharp permutationp-groups

A permutation groupG of finite degreed is called a sharp permutation group of type {k},k a non-negative integer, if every non-identity element ofG hask fixed points and |G|=d−k. We characterize sharp non-abelianp-groups of type {k} for allk.

Graded identities forT-prime algebras over fields of positive characteristic

In this paper we study 2-graded polynomial identities. We describe bases of these identities satisfied by the matrix algebra of order twoM 2(K), by the algebraM 1,1(G), and by the algebraG ⊗ K G. HereK is an arbitrary infinite field of characteristic not 2,G stands for the Grassmann (or exterior) algebra of an infinite dimensional vector space overK, andM 1,1(G) is the algebra of all 2×2 matrices overG whose entries on the main diagonal are even elements ofG, and those on the second diagonal are odd elements ofG. The gradings on these three algebras are supposed to be the standard ones.

Random generation of finite simple groups byp-regular orp-singular elements

Letp be a fixed prime and letG be a finite simple group. It is shown that two randomly chosen elements ofG of orders prime top generateG with probability tending to 1 as |G| → ∞. This answers a question of Kantor. Some related results are also established.

A technique for constructing divisible difference sets

An (m, n, k, λ1,λ2) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy−1:x, y e D} contains exactly λ1 copies of each nonidentity element ofN and exactly λ2 copies of each element ofG N. It is called semi-regular ifk > λ1 and k2=mnλ2. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.

Free Products of Units in Algebras I. Quaternion Algebras

LetAbe a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units ofAto generate a free group. Using the well-known isomorphism betweenSO(3,R) and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A consequence is the following: letkGbe the group algebra of a residually (torsion-free nilpotent) groupGover a fieldkwhose characteristic is not 2. Ifxandyare any pair of noncommuting elements ofG, andc,d∈k* then 1+cxand 1+dygenerate a free subgroup of the Malcev–Neumann field of fractions ofkG.

Swan Modules and Realisable Classes for Kummer Extensions of Prime Degree

Letl>2 be a prime number. Let OFdenote a ring of algebraic integers of a number fieldFand letGbe the cyclic group of orderl. Consider the ring of integers OLas a locally free OK[G]-module whereL/Kis a tamely ramified Galois extension of number fields with Galois group isomorphic toG. The classes of rings of integers obtained in such a way for a fixed number fieldKcontaining thelth roots of unity form the subgroup of realisable classesRin the locally free class groupCl(OK[G]). On the other hand, one may also look at the Swan subgroupTofCl(OK[G]), formed by the classes of locally free OK[G]-ideals (s,Σ) wheresin OKis relatively prime toland Σ denotes the sum of the elements ofG. We show thatT(l−1)/2⊂R∩D, whereDis the kernel group ofCl(OK[G]). We also determine necessary and sufficient conditions for when a realisable class is a Swan class. Last, we show thatR∩Dis nontrivial forK=Q(ζl) whenl>3 by providing a nontrivial lower bound for the size of the Swan subgroupT.

The diameters of almost all Cayley digraphs

LetG be a finite group of ordern andS be a subset ofG not containing the identity element ofG. Letp (0<p<1) be a fixed number. We define the set of all labelled Cayley digraphsX(G,S) (S<-G\{1}) ofG as a sample space and assign a probability measure by requiringP(aeS)=p for anya∈G\{1}. Here it is shown that the probability of the set of Cayley digraphs ofG with diameter 2 approaches 1 as the ordern ofG approaches infinity.

Some Algorithms for Nilpotent Permutation Groups

LetG,HandEbe subgroups of a finite nilpotent permutation group of degreen. We describe the theory and implementation of an algorithm to compute the normalizerNG(H) in time polynomial inn, and we give a modified algorithm to determine whetherHandEare conjugate underGand, if so, to find a conjugating element ofG. Other algorithms produce the intersectionG∩Hand the centralizerCG(H). The underlying method uses the imprimitivity structure of 〈G,H〉 and an associated canonical chief series to reduce computation to linear operations. Implementations in GAP and Magma are practical for degrees large enough to present difficulties for general-purpose methods.

Differences of functions and groups generators for compact Abelian groups

LetG be a Hausdorff compact Abelian group andC be the component of the identity element ofG. We consider a special class, ℋ(G), of functions inL2(G) whose Fourier series satisfy certain convergence conditions (stronger than absolute convergence). We show thatG/C is topologically generated by not more thann elements if and only if, for each functionf in ℋ(G), there area1,...,an inG and functionf1,...fn in ℋ(G) such that $$f = \sum\limits_{j = 1}^n {(f_j - \delta _{aj} * f_j ),}$$ where * is convolution defined in the usual sense, and δa denotes the Dirac measure ataeG.

Infinite dimensional groups and Riemann surface field theories

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.

Covering abelian groups with cyclic subsets

Letk andm be positive integers. An abelian groupG is said to have ann-cover if there is a subsetS ofG consisting ofn elements such that every non-zero element ofG can be expressed in the formig for some elementg inS and integeri, 1 ≤i ≤ k. Lets n (k) be the largest order of abelian groups that have ann-cover. We investigate the behavior ofs n (k)/k ask → ∞ andn is fixed.

Consistency of a ?-theory withn-tuples and easy term

We give here a model-theoretical solution to the problem, raised by J.L: Krivine, of the consistency of λβη+U(G)+Ω=t, wheret is an arbitrary λ-term,G an arbitrary finite group of order, sayn, andU(G) the theory which expresses the existence of a surjectiven-tuple notion, such that each element ofG behaves simultaneously as a permutation of the components of then-tuple and as an automorphism of the model. This provides in particular a semantic proof of the βη-easiness of the λ-term Ω.

Some remarks on profinite HNN extensions

We extend a construction of Higman, Neumann and Neumann [LS, IV.3.1] and show that every profinite groupG with only countably many open subgroups embeds in a 2-generated profinite groupE in which all torsion elements are conjugate to elements ofG; ifG is pro-p,E can be chosen pro-p. This answers a question of Wilson (oral communication) and generalises a result of Lubotzky and Wilson [LW].

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC G(a) for 1≠a ∈A.

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X 2|<|X|2, whereX 2={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism.