We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply these formulas effectively, the parameter c, which in general is hard to control, is studied in some important situations. These results are then used to provid...

suppose $g$ is a finite group, $a$ and $b$ are conjugacy classes of $g$ and $eta(ab)$ denotes the number of conjugacy classes contained in $ab$. the set of all $eta(ab)$ such that $a, b$ run over conjugacy classes of $g$ is denoted by $eta(g)$.the aim of this paper is to compute $eta(g)$, $g in { d_{2n}, t_{4n}, u_{6n}, v_{8n}, sd_{8n}}$ or $g$ is a decomposable group of order $2pq$, a group of...

We describe a set of subgroups isomorphic to S4 in PΓO(5, q), q ≡ 5(6), and prove that they belong to exactly two different conjugacy classes. As an application we use a representative group in each of the conjugacy classes to construct a number of BLT-sets of Q(4, q), some of which were previously found by computer in [5] (see also [4]), others of which are new.

A new general formula for calculation of number of conjugacy classes of subgroups of given index in a finitely generated group is obtained.

We give another proof of an observation of Thévenaz [4] and present a fusion system version of it. Namely, for a saturated fusion system F on a finite p-group S, we show that the number of the F-conjugacy classes of cyclic subgroups of S is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realiz...

‎There are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{Z}_{2}$ or $mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this...