On the product of element orders of some finite groups.

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]