Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order

Abstract In a finite group G, let π e ⁢ ( G ) {\pi_{e}(G)} be the set of orders of elements of G, let s k {s_{k}} denote the number of elements of order k in G, for each k ∈ π e ⁢ ( G ) {k\in\pi_{e}(G)} , and then let nse ⁡ ( G ) {\operatorname{nse}(G)} be the unordered set { s k : k ∈ π e ⁢ ( G ) } {\{s_{k}:k\in\pi_{e}(G)\}} . In this paper, it is shown that if | G | = | L 2 ⁢ ( q ) | {\lvert G\rvert=\lvert L_{2}(q)\rvert} and nse ⁡ ( G ) = nse ⁡ ( L 2 ⁢ ( q ) ) {\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))} for some prime-power q, then G is isomorphic to L 2 ⁢ ( q ) {L_{2}(q)} .