Related papers: The conjugacy class number k(G) - a different pers…

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K is normal in M, then K^G\cap M=K, where K^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every…

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie…

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the…

Let $\F$ be a field with a non-trivial involution $c: \alpha \to \alpha^c$. An element $g \in {\rm GL}_n(\F)$ is called $c$-real if it is conjugate to $(g^c)^{-1}$. We prove that for $n \geq 2$, $g \in {\rm GL}_n(\F)$ is $c$-real if and…

A common fixed point property for semigroups is applied to show that the group algebra $L^1(G)$ of a locally compact group $G$ is $2m$-weakly amenable for each integer $m\geq 1$.

The minimal faithful permutation degree of a finite group $G$, denote by $\mu(G)$ is the least non-negative integer $n$ such that $G$ embeds inside the symmetric group $\Sym(n)$. In this paper, we outline a Magma proof that 10 is the…

Let $G$ be an algebraic group over a field $k$. We call $g\in G(k)$ {\bf real} if $g$ is conjugate to $g^{-1}$ in $G(k)$. In this paper we study reality for groups of type $G_2$ over fields of characteristic different from 2. Let $G$ be…

It is shown that for the conjugation action of the symmetric group $S_n,$ when $n=6$ or $n\geq 8,$ all $S_n$-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition $\lambda$ of $n$ with at least…

Let $\pi$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class…

We outline how the group $K_{-1}( \mathbb{Z}[G] )$ for a finite group $G$ can be computed using the computer language $GAP$ and compile a table of all groups $G$ of order less than $100$ that have torsion in $K_{-1}( \mathbb{Z}[G] )$.

In this paper we find the number of conjugate $\pi$-Hall subgroups in all finite almost simple groups. We also complete the classification of $\pi$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.

Let $G$ be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic $p \ge 0$. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if whenever $H$ is contained in a parabolic…

In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n is greater or equal to 3 and alternating group A_n, when n is greater or equal to 4. It turns out that the…

For any subgroup H of Out(F_n), either H has a finite index subgroup that fixes the conjugacy class of some proper, nontrivial free factor of F_n, or H contains a fully irreducible element phi, meaning that no positive power of phi fixes…

We obtain formulae for the numbers of isomorphism and conjugacy classes of non-identity proper subgroups of the groups $G={\rm PSL}_2(p)$, $p$ prime, and for the numbers of those conjugacy classes which do or do not consist of…

In this paper, we prove that if G is a finite simple group with the same-size conjugacy class set U(G) = U(PSL(2, 11)), then G is isomorphic to PSL(2, 11).

In this note we give some new results concerning the subgroup commutativity degree of a finite group $G$. These are obtained by considering the minimum of subgroup commutativity degrees of all sections of $G$.

We determine the range of the des statistic on every conjugacy class in the symmetric group $S_n$, prove that the minimum is $1$ (except for the identity class), and show that every intermediate value from $1$ to the maximum value is…

For a finite group $G,$ we define the concept of $G$-partial permutation and use it to show that the structure coefficients of the center of the wreath product $G\wr \mathcal{S}_n$ algebra are polynomials in $n$ with non-negative integer…

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep…