Bounds on number of conjugacy classes in terms of number of elements of a group ?
There is a theorem of E. Landau which proves that if you fix a positive integer h, there are only finitely many finite groups with h conjugacy classes. This proof is more number theory than group theory, in fact. More recently, one person who has worked more extensively on this question using more group theory is L. Pyber.
See MR1182481 (93i:20028) Pyber, L., Finite groups have many conjugacy classes, J. London Math. Soc. (2) 46 (1992), no. 2, 239–249. From the review by I. Ya. Subbotin:
Let $k(G)$ denote the number of conjugacy classes of a finite group $G$. R. Brauer observed that for every group $G$ of order $n$ we have $k(G)\ge\log\log n$, and proposed the problem of finding substantially better bounds [R. Brauer, in Lectures on modern mathematics, Vol. 1, 133--175, Wiley, New York, 1963; MR0178056 (31 #2314)]. The author proves that every group of order $n$ contains at least $\epsilon\log n/(\log\log n)^8$ conjugacy classes for some fixed $\epsilon$. This essentially settles the problem of Brauer.
Zbl 1227.20014. of A. Jaikin-Zapirain paper says that there is conjecture by Ya. Berkovich and V. Zhmud that: number of conj classes > number of prime factos of G (achieved for M_22 and PSL(3,4) )
Quote from Andrei Jaikin-Zapirain paper (Adv. Math. 227, No. 3, 1129-1143 (2011). )
Conjecture. There exists a constant C> 0 such that any finite group G of order n satisfies k(G) > C log2( n).
Main theorem of this paper is the following:
In this paper we establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group.
Theorem 1.1. There exists a (explicitly computable) constant C > 0 such that every finite nilpotent group G of order n 8 satisfies
k(G) > C log2 (n) ((log2 log2 n) / ( log2 log2 log2 n) )
Introduction to the paper contains discussion of some history of the subject is quite readable.
On p-groups having the minimal number of conjugacy classes of maximal size A. Jaikin-Zapirain, M. F. Newman and E. A. O’Brien
A long-standing question is the following: do there exist p-groups of odd order having precisely p-1 conjugacy classes of the largest possible size? We exhibit a 3-group with this property.
http://springerlink.com/content/n737pq17655582qv 1970 The number of conjugacy classes in a finite group. Patrick X. Gallagher.
This paper contains results comparing number of conjugacy classes in a group and in its subgroup.
http://arxiv.org/abs/1102.4107
Multiplicities of conjugacy class sizes of finite groups
Hung Ngoc Nguyen
It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Zaikin-Zapirain. In this note, we prove that if $G$ is a finite simple group then the order of $G$, denoted by $|G|$, is bounded in terms of the largest multiplicity of its conjugacy class sizes and that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group $G$ is $m$, then $|G|$ is bounded in terms of $m$.