For a group G and X a subset of G the commuting graph of G on X, denoted by C(G,X), is the graph whose vertex set is X with x, y ∈ X joined by an edge if x 6= y and x and y commute. If the elements in X are involutions, then C(G,X) is called a commuting involution graph. This paper studies C(G,X) when G is a 3-dimensional projective special unitary group and X a G-conjugacy class of involutions...

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group Sn are all conjugate. In this paper, we study possible generalizations of this property to other types of Coxeter groups. We show that Schützenberger’s result is a special case of a general result on “smooth” two-sided cells. Furthermore, we consider Kottwitz’s conjecture concerning th...

Suppose that G is a group, X a subset of G and π a set of natural numbers. The π-product graph Pπ(G,X) has X as its vertex set and distinct vertices are joined by an edge if the order of their product is in π. If X is a set of involutions, then Pπ(G,X) is called a π-product involution graph. In this paper we study the connectivity and diameters of Pπ(G,X) when G is a finite symmetric group and ...

Let M K be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S × S of spheres. We show that a finite group of odd order acts freely on M K if and only if it acts freely on S × S. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k 6= 2 − 1, then M K does not admit any free TOP in...

We define the orthogonal sum of quadratic pairs and we show that there is no Witt cancellation theorem for this operation in characteristic 2. 1. Introduction. Quadratic pairs on central simple algebras were defined in [5]. They play the same role for quadratic forms as involutions for symmetric or skew-symmetric bilinear forms. In particular, they can be used to define twisted orthogonal group...

The aim of this article is to provide a characterization of quadratic forms of low dimension such that the canonical involutions of the their Clifford algebras are hyperbolic.

Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated. 2000 Mathematics Subject Classification: 16W10, 11E39

We give an explicit construction of sharply 2-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.