A $p$-nilpotency criterion for finite groups (Q2416457)

All groups below are finite. Following \textit{M. Tǎrnǎuceanu} [Acta Math. Hung. 155, No. 2, 499--501 (2018; Zbl 1449.20017)], the authors denote by $\varphi(G)$ the number of elements of a group $G$ of order the exponent of $G$. The authors prove that for a prime $p$ and a group $G$, the group $G$ is $p$-nilpotent if and only $\varphi(K) \not = 0$ for every section $K$ of $G$ satisfying $O_{p'}(K) = \langle 1\rangle$. In fact, they also show that only certain of these $p'$-reduced sections $K$ need to be considered. They then derive as a corollary of their theorem the following result of Tǎrnǎuceanu [loc. cit.]: the group $G$ is nilpotent if and only if $\varphi(K) \not = 0$ for every section $K$ of $G$.