Sums of element orders in groups of order 2m with m odd

Abstract

If G is a finite group, then ψ(G) denotes the sum of orders of all elements of G and if k is a positive integer, then Ck denotes a cyclic group of order k. Moreover, ψ(Ck) will be sometimes denoted by ψ(k). In this article we deal with groups of order with m odd. Our main results are the following two theorems: Theorem 7. Let G be a non-cyclic group of order , with m an odd integer. Then . Moreover, if and only if , where with and S3 is the symmetric group on three letters. Theorem 8. Let Δn be the set of non-cyclic groups of the fixed order , where m is an odd integer, and suppose that , where pi are distinct primes and αi are positive integers for all i. If , then , where . Moreover, satisfies if and only if , where is the dihedral group of order 2l.