On the sum of non-cyclic subgroups order in a finite group.

Let G be a finite group and δ (G) = 1 | G | ∑ H ≤ G { | H | | H is non-cyclic}. In this paper, we show that some arithmetical conditions of δ (G) influence the structure of G. Firstly, we prove that if δ (G) < 13 3 , then G is solvable. Secondly, we determine the structure of finite groups with δ (G) ≤ 2 . Moreover, we prove that if δ (G) < 1 + 4 | G | , then G is supersolvable, and we also determine the structure of finite groups G with δ (G) = 1 + 4 | G | . Finally, we show that δ (G) < c does not imply the supersolvability of G for any constant c ∈ (1 , ∞) . [ABSTRACT FROM AUTHOR]