Related papers: Calculating conjugacy classes in Sylow $p$-subgrou…

H\'ethelyi and K\"ulshammer showed that the number of conjugacy classes $k(G)$ of any solvable finite group $G$ whose order is divisible by the square of a prime $p$ is at least $(49p+1)/60$. Here an asymptotic generalization of this result…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

We find an upper bound for the number of groups of order $n$ up to isomorphism in the variety $G = A_pA_qA_r$, where $p$, $q$ and $r$ are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of…

To classify the finite dimensional pointed Hopf algebras with Weyl group $G$ of $E_6$, $E_7$, $F_4$, $G_2$, we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means…

We construct explicit quantization of closed conjugacy classes of the complex symplectic group SP(2n) with non-Levi isotropy subgroups through an operator realization on highest weight modules over the quantum group U_q(sp(2n)).

Let $k(G)$ be the number of conjugacy classes of finite groups $G$ and $\pi_e(G)$ be the set of the orders of elements in $G$. Then there exists a non-negative integer $k$ such that $k(G)=|\pi_e(G)|+k$. We call such groups to be $co(k)$…

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which…

A classical conjecture by Graham Higman states that the number of conjugacy classes of $U_n(q)$, the group of upper triangular $n\times n$ matrices over $\mathbb{F}_q$, is polynomial in $q$, for all $n$. In this paper we present both…

For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.

In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.

We give an efficient algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie…

Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$: 1. List a representative for each conjugacy class of $G$. 2. Given $x \in G$, describe the centralizer of…

We construct explicit quantization of semisimple conjugacy classes of the complex orthogonal group SO(N) with non-Levi isotropy subgroups through an operator realization on highest weight modules of the quantum group U_q(so(N)).

We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…

We give a general construction for right conjugacy closed loops, using $GL(2,q)$ for $q$ a prime power. Under certain conditions, the loops constructed are simple, giving the first general construction for finite, simple right conjugacy…

We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…

Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected…

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…

Suppose $G$ is a connected complex semisimple group and $W$ is its Weyl group. The lifting of an element of $W$ to $G$ is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of $W$ to the set of semisimple…

We classify, up to conjugacy, the finite subgroups of PGL(2,K) of order prime to char(K).