Abstract
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 , ∞ ) .
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