For a finite group $G$ let $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$. We give a short proof of a theorem of Brandl, which classifies finite groups with $nu(G)=1$.

‎Let $G$ be a finite group and $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$‎. ‎In this paper‎, ‎all nilpotent groups $G$ with $nu(G)=3$ are classified‎.  

We give a characterization of the group property of being with infinite conjugacy classes (or icc, i.e. 6= 1 and of which all conjugacy classes beside 1 are infinite) for split extensions of group.

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group in...

We classify the real and strongly real conjugacy classes in GLnðqÞ and SLnðqÞ. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This paper is the first of two that together classify the real and strongly real classes in GLnðqÞ, SLnðqÞ, PGLnðqÞ, PSLnðqÞ, and all quasi-simple covers of PSLnðqÞ.

We classify the real and strongly real conjugacy classes in GLnðqÞ and SLnðqÞ. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This paper is the first of two that together classify the real and strongly real classes in GLnðqÞ, SLnðqÞ, PGLnðqÞ, PSLnðqÞ, and all quasi-simple covers of PSLnðqÞ.

Let G be a real compact connected simple Lie group, and g its Lie algebra. We study the problem of determining, from root data, when a sum of adjoint orbits in g, or a product of conjugacy classes in G, contains an open set. Our general methods allow us to determine exactly which sums of adjoint orbits in su(m) and products of conjugacy classes in SU(m) contain an open set, in terms of the high...

In [8, §8], the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow p-subgroup U(q) of a finite Chevalley group G(q), valid when q is a power of a good prime for G(q). In this paper we develop this algorithm and discuss an implementation in the computer algebra language GAP. Using the resulting computer program we are able to calculate the pa...

Let G ⊂ SL(n, C) be a finite subgroup and φ:Y → X = C/G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with “junior” conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with KY = 0 ...

Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2(R) to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondenc...