The number of distinct element orders in a finite group

For a finite group $G$ let $\omega (G)$ denote its spectrum, the set of orders of its elements, and set $\eta (G)=\mathrm{\mid }\omega (G)\mathrm{\mid }$, the number of distinct element orders. The set $\omega (G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $\eta$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+{\sum }_{p\mid \mathrm{\mid }G\mathrm{\mid }}{v}_p(\exp G)\le \eta (G)\le \tau (\exp G)$, in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving $\eta (G)=\tau (\exp G)$. We also prove $\eta (G)\le \tau (\mathrm{\mid }G\mathrm{\mid })$ with equality only for cyclic groups, and $\eta (G)\ge \pi (\mathrm{\mid }G\mathrm{\mid })+1$ with equality only when every nontrivial element has prime order. We show that $\eta (G)\le 3$ forces solvability by an argument resting only on Burnside’s theorem, and, combining this with the recognition of $A_5$ by its spectrum, that $A_5$ is the unique non-solvable group with $\eta (G)=4$. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of $\eta$ over all groups of a fixed order.