Cinarcı–Keller conjecture on the minimum number of conjugacy classes

Let $p$ be a prime. Let $a$ and $b$ be positive integers such that

$$p-1=ab$$

and such that $|a-b|$ is minimal. For a finite group $G$, let $k(G)$ denote the number of conjugacy classes of $G$. Cinarcı–Keller conjecture. If $p$ divides $|G|$, then

$$k(G)\geq a+b,$$

and equality holds if and only if

$$G\cong \mathsf{C}_p\rtimes\mathsf{C}_a\quad\text{or}\quad G\cong \mathsf{C}_p\rtimes\mathsf{C}_b.$$

Here the two semidirect products are the groups occurring in the source's equality characterization. The conjecture is presented as an open problem; the paper notes that its lower bound is supported by the authors' main theorem in cases where the relevant integer belongs to the specified divisor set.