On CLT and non-CLT groups

Abstract

In this note, we prove that for every integer $d\ge 2$ which is not a prime power, there exists a finite solvable group $G$ such that $d\mid |G|$, $\pi (G)=\pi (d)$ and $G$ has no subgroup of order $d$. We also introduce the CLT-degree of a finite group and answer two questions about it.

Accepted for publication in Ann. Mat. Pura Appl