On the widths of finite groups (I)

Let $G$ be a finite group, $\pi (G)$ be the set of prime divisors dividing the order of $G$ and ${\pi }_e(G)$ (spectrum) denote the set of element orders of $G$. We define $w_o(G)$ = $\mathrm{\mid }\pi (G)\mathrm{\mid }$ the width of order of $G$ and $w_s(G)$ = max$\{\mathrm{\mid }\pi (k)\mathrm{\mid }\mathrm{\mid }k\in {\pi }_e(G)\}$ the width of spectrum of $G$. In this paper, we discuss the cases of $w_o(G)$ and $w_s(G)$ are small, prove several new results and give a survey about the two widths of groups.