Conjugacy Classes Of Involutions Research Articles

Lengths of involutions in finite Coxeter groups

Let W be a finite Coxeter group and X a subset of W . The length polynomial L W , X ( t ) is defined by L W , X ( t ) = ∑ x ∈ X t ℓ ( x ) , where ℓ is the length function on W . If X = { x ∈ W : x 2 = 1 } then we call L W , X ( t ) the involution length polynomial of W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W . In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type B n and D n . Moreover, we give a counterexample to a unimodality conjecture stated in [11] .

On construction of certain Fischer embedded subgroups generated by 3-transpositions in Fi22

In this paper, we investigate the Fischer group [Formula: see text]. This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups [Formula: see text] and [Formula: see text], Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results: • The rank of the groups [Formula: see text], the Weyl group W of type [Formula: see text] over fields of characteristic 2, the group M of shape [Formula: see text] and the group [Formula: see text] are computed. If G is a 3-transposition group (Fischer embedded) for a class [Formula: see text] of transpositions, then [Formula: see text] is defined as [Formula: see text], all elements in X commute[Formula: see text]. • We prove that the subgroups [Formula: see text] and [Formula: see text] are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is [Formula: see text] and [Formula: see text] where G is the group generated by a proper subset E of D, [Formula: see text] is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.

Commuting involution graphs in classical affine Weyl groups

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let W be a classical Weyl group of rank n, with its corresponding affine Weyl group. Our main result is that if X is a conjugacy class of involutions in then the commuting involution graph is either disconnected or has diameter at most n + 2. This bound is known to hold for types and so the main work of this paper is to prove the theorem for types and

On the structure of generalized symmetric spaces of SLn𝔽q)

ABSTRACTIn this paper we extend previous results regarding SL2(k) over any finite field k by investigating the structure of the symmetric spaces for the family of special linear groups SLn(k) for any integer n>2. Specifically, we discuss the generalized and extended symmetric spaces of SLn(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal.

Linear idempotents in Matsuo algebras

Matsuo algebras are an algebraic incarnation of 3-transposition groups with a parameter $\alpha$, where idempotents takes the role of the transpositions. We show that a large class of idempotents in Matsuo algebras satisfy the Seress property, making these nonassociative algebras well-behaved analogously to associative algebras, Jordan algebras and vertex (operator) algebras. We calculate eigenvalues in the Matsuo algebra of ${\rm Sym}(n)$ for any $\alpha$, generalising some vertex algebra results for which $\alpha=\frac{1}{4}$. Finally, in the Matsuo algebra of the root system ${\rm D}_n$, we show $n-3$ conjugacy classes of involutions coming from the Weyl group are in natural bijection with idempotents in the algebra via their fusion rules.

Local Fusion Graphs and Sporadic Simple Groups

For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.

On the ranks of Fi22

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G:X) to be the minimum number of elements of X generating G. Moori investigated rank(Fi22:X), where X is a conjugacy class of involutions in the Fischer group Fi22. In the present article, we completely determine the ranks of the Fischer group Fi22 by computing rank(Fi22:X), where X is any conjugacy class of Fi22.

Involutions in Janko’s simple group J4

AbstractIn this paper we determine the suborbits of Janko’s largest simple group in its conjugation action on each of its two conjugacy classes of involutions. We also provide matrix representatives of these suborbits in an accompanying computer file. These representatives are used to investigate a commuting involution graph for J4.Supplementary materials are available with this article.

Commuting Involution Graphs for $3$-Dimensional Unitary Groups

For a group $G$ and $X$ a subset of $G$ the commuting graph of $G$ on $X$, denoted by $\cal{C}(G,X)$, is the graph whose vertex set is $X$ with $x,y\in X$ joined by an edge if $x\neq y$ and $x$ and $y$ commute. If the elements in $X$ are involutions, then $\cal{C}(G,X)$ is called a commuting involution graph. This paper studies $\cal{C}(G,X)$ when $G$ is a 3-dimensional projective special unitary group and $X$ a $G$-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

Combinatorial method in the coset enumeration of symmetrically generated groups

Several finite groups, including all non-abelian finite simple groups, can be generated by finite conjugacy classes of involutions. We describe a coset enumeration algorithm that is particularly developed for a group defined in this manner. We illustrate the process by using some rather small examples for which the enumerations can be worked manually.

Automorphisms of Coxeter groups of type K n

A Coxeter system (W, S) is said to be of type Kn if the associated Coxeter graph ΓS is complete on n vertices and has only odd edge labels. If W satisfies either of: (1) n = 3; (2) W is rigid; then the automorphism group of W is generated by the inner automorphisms of W and any automorphisms induced by ΓS. Indeed, Aut(W) is the semidirect product of Inn(W) and the group of diagram automorphisms, and furthermore W is strongly rigid. We also show that if W is a Coxeter group of type Kn then W has exactly one conjugacy class of involutions and hence Aut(W) = Spec(W).

A characterization of the full wreath product

A groupoid [3, 17] is a set Q endowed with a binary product, that is, a map from Q × Q to Q. In his 1964 paper [7], Bernd Fischer studied distributive quasigroups, which by definition are groupoids Q for which right multiplication by any fixed element gives an automorphism of Q as does left multiplication. Fischer proved that the right multiplication group R(Q) of a finite distributive quasigroup Q is solvable. He did this by showing that, for a minimal counterexample, the right multiplications T = {μa : g 7→ ga | a ∈ Q } are a generating conjugacy class of involutions in R(Q) ≤ Aut(Q) with the additional property that |tr| = 3 for distinct t and r from T . He then proved that this property forces finite R(Q) to have a normal 3-group of index 2. This led Fischer to consider [8, 9, 10] the extent to which finite symmetric groups can be characterized through being generated by a conjugacy class of involutions with all products of order 1, 2, or 3—a class of 3-transpositions, since the model is the transposition (2-cycle) class of Sym(Ω), the symmetric group on the set Ω. In a landmark theorem [10], Fischer found all finite 3transposition groups with no nontrivial solvable normal subgroups, discovering three new sporadic simple groups along the way. At the same time that Fischer was considering distributive quasigroups, George Glauberman was working on certain special groupoids, called Bruck loops. Glauberman [13] proved that finite Bruck and finite Moufang loops of odd order are solvable. His approach was similar to Fischer’s. He constructed a canonical conjugacy class T of involutory loop permutations with the additional property that |tr| was always odd for t and r from T . In his famous Z∗-theorem [14], Glauberman then proved that a finite group generated by such a class T has a normal subgroup of odd order and index 2 (a result also proved by Fischer [7] in the special case where all orders |tr| are powers of some fixed odd prime). Fischer’s and Glauberman’s work on finite quasigroups and loops had a profound effect on the theory of finite simple groups. For a normal set of involutions

Conjugacy classes of involutions in the Lorentz group $Omega;(V) and in SO(V)

The Lorentz group Ω(V) is bireflectional and all involutions in Ω(V) are conjugate. More generally, we give conditions for two involutions to be conjugate in SO ( V ), provided that V is a vector space over a finite field or over an ordered field.

Conjugacy classes of involutions in the Lorentz group [formula omitted] and in SO( V)

The Lorentz group Ω(V) is bireflectional and all involutions in Ω(V) are conjugate. More generally, we give conditions for two involutions to be conjugate in SO( V), provided that V is a vector space over a finite field or over an ordered field.

Involutions in Conway's Largest Simple Group

AbstractIn this paper, the authors determine the suborbits of Conways largest simple group in its conjugation action on each of its three conjugacy classes of involutions. Matrix representatives of these suborbits are also provided in an accompanying computer file.

Minimal and Maximal Length Involutions in Finite Coxeter Groups

Let W be a finite irreducible Coxeter group. The aim of this paper is to describe the maximal and minimal length elements in conjugacy classes of involutions in W.

On automorphisms of free pro-𝑝-groups. I

Let F F be a (topologically) finitely generated free pro- p p -group, and β \beta an automorphism of F F . If p ≠ 2 p \ne 2 and the order of β \beta is 2, then there is some basis of F F such that β \beta either fixes or inverts its elements. If p p does not divide the order of β \beta , then the subgroup of F F of all elements fixed by β \beta is (topologically) infinitely generated; however this is not always the case if p p divides the order of β \beta .

Even automizers and conjugacy of involutions

The series “Weakly closed p-elements and odd automizers” [l IA-1 1 C] dealt with groups G in which a Sylow p-normalizer is “essentially” an odd-order Frobenius group with a non-trivial Schur multiplier. This sequel explores the analogous situation in which the Frobenius group has an even-order complement; in this case the desired conclusion is not p-length 1, but rather a restriction on the number of conjugacy classes of involutions in G. The main result is:

Characterization of a family of simple groups by their character table

AbstractThe paper establishes a method for bounding the 2-rank of a simple group with one conjugacy class of involutions, by means of its character table. For many groups of 2-rank ≦ 4, this bound is shown to be exact. The main result is that the simple groups G2(q),(q,6) = 1, are characterized bv their character table.

Conjugacy in groups with dihedral 3-normalisers

Much work has been carried out on the classification of finite simple groups in terms of the structures of centralisers of involutions. However, it is sometimes the case that these classification results cannot be applied to particular problems even although information is available about one conjugacy class of involutions. The trouble is that information about the other classes can be almost non-existent. In this paper we deal with a situation where character theory can be employed to give a strong connection between the orders of centralisers of different classes of involutions, enabling information about one class to be used to give information about other classes. We prove the following result.