Two new criteria for solvability of finite groups

A density result on the sum of element orders of a finite group

“…The author conjectures that if ψ ′ (G) > 31 77 , then G is supersolvable, this result being completely proved by Baniasad Azad and Khosravi in [5]. Finally, in [7], Herzog, Longobardi and Maj established that once ψ ′ (G) > 1 6.68 , G is solvable. They outlined the fact that their result may be further improved by replacing the lower bound 1 6.68 with 211 1617 .…”

A result on the sum of element orders of a finite group

“…It is proved that if G is a group of order n and ψ(G) > 211 1617 ψ(C n ), then G is solvable (see [2], [7]). Baniasad Azad and Khosravi in [2] gave some other groups the equality holds for.…”

“…Here, we prove that these groups are the only groups the equality holds for. Thus, we prove a modified version of Conjecture 6 in [7] as follows.…”

“…Therefore by the non-solvability of G/H, [ x : H] = 12 and |G/H| = 168. It follows that G/H ∼ =L 2(7) and ψ(G/H) = 715.…”

A criterion for solvability of a finite group by the sum of element orders

“…We can see that ψ(C p α ) = p 2α+1 +1 p+1 , where α ∈ N. In [5], an exact upper bound for sums of element orders in non-cyclic finite groups is given. In [6], the authors give two new criteria for solvability of finite groups. They proved that, if G is a group of order n and ψ(G) ≥ ψ(C n )/6.68, then G is solvable.…”