The number of distinct element orders in a finite group

Abstract

For a finite group $G$ let $ω(G)$ denote its spectrum, the set of orders of its elements, and set $η(G)=|ω(G)|$, the number of distinct element orders. The set $ω(G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $η$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+\sum_{p\mid|G|}v_p(\exp G)\leη(G)\leτ(\exp G)$, 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 $η(G)=τ(\exp G)$. We also prove $η(G)\leτ(|G|)$ with equality only for cyclic groups, and $η(G)\geπ(|G|)+1$ with equality only when every nontrivial element has prime order. We show that $η(G)\le3$ 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 $η(G)=4$. 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 $η$ over all groups of a fixed order.