A finite group G satisfies |H|-|K| divides ψ(H)-ψ(K) for all subgroups K ≤ H if and only if G is a p-group of exponent p.
Amiri,On a bijection between a finite group and cyclic group, J
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On a divisibility condition related to the sum of element orders of a finite group
A finite group G satisfies |H|-|K| divides ψ(H)-ψ(K) for all subgroups K ≤ H if and only if G is a p-group of exponent p.