We study the commuting graph on elements of odd prime order in finite simple groups. The results are used in a forthcoming paper describing the structure of Bruck loops and Bol loops of exponent 2.

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V , where K is a field of characteristic p ≥ 0. If F = K(t) is the field of rational functions on the projective line P1/K , consider the K(t)-endomorphism A = X+ tY of V . If p = 0, or if A = 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V ). For all geometric points (a : b) o...

In this paper the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p) is studied, where p is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space Rep(Z, Gm,p) of flat principal Gm,p–bundles over the n–torus is completely described for all values of...

By normalizing a component of the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkähler orbifolds which satisfy the conditions to be mirror partners in the sense of Strominger-Yau-Zaslow. The same holds true for commuting quadruples in a compact Lie group. The Hodge numbers of the mirror partn...

The purpose of this paper is to introduce a new method of “stabilizing” spaces of homomorphisms Hom(π,G) where π is a certain choice of finitely generated group and G is a compact Lie group. The main results apply to the space of all ordered n-tuples of pairwise commuting elements in a compact Lie group G, denoted Hom(Zn, G), by assembling these spaces into a single space for all n ≥ 0. The res...

Let $G$ be a non-abelian group. The non-commuting graph $Gamma_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joined if and only if they do not commute.In this paper we study some properties of $Gamma_G$ and introduce $n$-regular $AC$-groups. Also we then obtain a formula for Szeged index of $Gamma_G$ in terms of $n$, $|Z(G)|$ and $|G|...

Let G be a connected reductive affine algebraic group defined over the complex numbers, and K ⊂ G be a maximal compact subgroup. Let X, Y be irreducible smooth complex projective varieties and f : X → Y an algebraic morphism, such that π1(Y ) is virtually nilpotent and the homomorphism f∗ : π1(X)→ π1(Y ) is surjective. Define R ( π1(X), G ) = {ρ ∈ Hom ( π1(X), G ) | A ◦ ρ factors through f∗} , ...

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the clas...

Let Γ denote the q-th stage of the descending central series of the free group on n generators Fn. For each q ≥ 2 and every topological group G, a simplicial space B∗(q, G) is constructed where Bn(q, G) = Hom(Fn/Γ , G) and the realizations B(q, G) = |B∗(q, G)| filter the classifying space BG. In particular for q = 2 this yields a single space B(2, G) assembled from all the n–tuples of commuting...

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding...