On the number of conjugacy classes of finite solvable groups of order divisible by $p^2$

Abstract

In 2003, Héthelyi and Külshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that $k(G)\geq (49p+7)/60$ for all primes $p$, which is sharp for $p=17$, and that $k(G)\geq (49p+451)/60$ for all primes $p>2719$, which is sharp for infinitely many primes $p$. Thus the slope $49/60$ found by Héthelyi and Külshammer is best possible, but the correct additive constant is $451/60$ rather than $1/60$.