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.
Key words: