Hence, it is easy to count the orbits of GL n q under the conjugation action of U n q but seems hard to do the same for the group U n q itself! We fix some further notation. Let V n q be the vector space of column vectors, a module for GL n q . Recall that a flag is a totally ordered set of n− 1 nonzero proper subspaces of V n q . For g in GL n q let f g be the number of flags fixed by g, so it...

A special type of conjugacy classes in symmetric groups is studied and used to answer a question about odd-degree irreducible characters.

Say that an element of a free group is a pure conjugate if it can be expressed from the generators using exclusively the conjugacy operation. We study free reductions in words representing pure conjugates. Using finite state automata, we attribute to the letters in such words levels that live in some free left distributive system. If a certain conjecture about this system is true, then reductio...

Let q be a power of a prime and n a positive integer. Let P (q) be a parabolic subgroup of the finite general linear group GLn(q). We show that the number of P (q)conjugacy classes in GLn(q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin in [1].

The original title for this essay was ‘What you always wanted to know about nilpotence but were afraid to ask’. I have changed it, because that title is not likely to be an accurate description of the current version. It is my attempt to understand and explain nilpotent conjugacy classes in the classical complex semi-simple Lie algebras (and therefore also, through the exponential map, of the u...

A group is said to have the R∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R∞ property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n ≥ 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point ...

Fullerene chemistry is nowadays a well-established field of both theoretical and experimental investigations‎. This study considers the symmetry of small fullerenes cage C24 and C28‎. ‎Using PM3 program for C24 and C28 fullerenes, Oh and Td symmetry were confirmed, respectively‎. ‎ The mentioned algorithm to compute the automorphism group of these fullerenes with connectivity and geometry of th...