For a finite group $G$ let $k(G)$ denote the number of conjugacy classes of $G$. Answering a question of Frobenius, Landau proved in 1903 that for a given $k$ there are only finitely many groups having $k$ conjugacy classes. Making this result explicit, we have $\log \log |G| < k(G)$ for any non-trivial finite group $G$. (The base of the logarithms will always be $2$ unless otherwise stated.) Problem $3$ of Brauer's list of problems is to give a substantially better lower bound for $k(G)$ than this.

Pyber proved that there exists a constant $\epsilon>0$ so that for every finite group $G$ of order at least $3$ we have $\epsilon \log |G|/{(\log \log |G|)}^{8} < k(G)$. Almost 20 years later Keller replaced the $8$ in the previous bound by $7$.

In our paper:

B. Baumeister, A. Maroti, H.P. Tong-Viet, Finite groups have more conjugacy classes, Forum Mathematicum, to appear.

we first give a further improvement to Pyber's theorem.

Theorem. For every $\epsilon > 0$ there exists a $\delta > 0$ so that for every finite group $G$ of order at least $3$ we have $\delta \log |G|/{(\log \log |G|)}^{3+\epsilon} < k(G)$.

There are many lower bounds for $k(G)$ in terms of $|G|$ for the various classes of finite groups $G$.
For example, Jaikin-Zapirain gave a better than logarithmic lower bound for $k(G)$ when $G$ is a nilpotent group. For supersolvable $G,$ Cartwright  showed $(3/5) \log |G| < k(G)$. For solvable groups the best bound to date is a bit worse than logarithmic and is due to Keller.

Conjecture. There is a universal constant $c > 0$ so that $c \log |G| < k(G)$ for any finite group $G.$

This conjecture has been intensively studied by many mathematicians including Bertram. Bertram observed that $k(G) = \lceil \log_3(|G|) \rceil$ when $G = \textrm{PSL}_3(4)$ or $\textrm{M}_{22}$ and checked the proposed bound for certain  small groups. He then proposed the following conjecture.

Conjecture. If $G$ is a finite group, then $\log_{3}|G| < k(G).$

In our paper, we answer Bertram's question in the affirmative for groups with a trivial solvable radical.

Theorem. Let $G$ be a finite group with a trivial solvable radical. Then $\log_{3} |G| < k(G)$.

Recall that  an element $x$ in a finite group $G$  is \emph{real} if $x$ and $x^{-1}$ are conjugate in $G.$ Equivalently, $x\in G$ is real if and only if $\chi(x)$ is real, that is, $\chi(x)\in \mathbb{R},$ for any $\chi\in\textrm{Irr}(G),$ where $\textrm{Irr}(G)$ denotes the set of all complex irreducible characters of $G.$ Of course, if $x\in G$ is real then every element in the conjugacy class $K:=x^G$ containing $x$ is also real and we say that $K$ is a \emph{real conjugacy class} and $|K|$ is a \emph{real class size}.

In my paper:

Tong-Viet, H. P. Groups with some arithmetic conditions on real class sizes. Acta Math. Hungar. 140 (2013), no. 1-2, 105–116.

I proved the following result.

Theorem. Let $G$ be a finite group. If the conjugacy class size of every odd prime power order real element in $G$ is a $2$-power or not divisible by $4,$ then $G$ is solvable.

We note that the analogous problem for complex irreducible characters does not hold as $4$ divides no irreducible complex character degrees of the alternating group of degree $7,$ but this group is clearly not solvable.

We now study in detail the structure of finite groups whose conjugacy class sizes of odd prime power order real elements are not divisible by $4.$ Note that $\textbf{O}^{2'}(G)$ is the smallest normal subgroup of $G$ such that the quotient $G/\textbf{O}^{2'}(G)$ is a group of odd order.

Theorem. Let $G$ be a finite group. If the conjugacy class size of every odd prime power order real element in $G$ is not divisible by $4,$ then $\textbf{O}^{2'}(G)$ is $2$-nilpotent and $G/\textbf{O}_{2'}(G)$ is $2$-closed.

If $n$ is a positive integer and $p$ is a prime, then we can write $n=n_pn_{p'},$ where $n_p$ is a power of $p$ and $n_{p'}$ is relative prime to $p.$ We will call $n_p$ the $p$-part of $n.$ One consequence of the previous theorem is that if the $2$-part of the conjugacy class sizes of all noncentral real elements of $G$ is exactly $2,$ then $\textbf{O}^{2'}(G)$ is $2$-nilpotent. To generalize this result, we  make the following conjecture.

Conjecture. Suppose that the sizes of all noncentral real conjugacy classes of a finite group $G$ have the same $2$-part. Then $\textbf{O}^{2'}(G)$ is $2$-nilpotent. In particular,  $G$ is solvable.

The group $\rm{PSL}_3(2)$ shows that we cannot restrict the hypothesis of this conjecture to noncentral odd order real conjugacy classes since this group has only one noncentral odd order real conjugacy class with size $7\cdot 2^3.$

One  application of this conjecture, if true, is a positive answer to a question due to G. Navarro:

Question.  If $|\textbf{C}_G(x)|=|\textbf{C}_G(y)|$ for all noncentral real elements $x$ and $y$ in $G,$ then $G$ is solvable.

In other words, a group $G$ is solvable if it has at most two real class sizes.