Non-supersolvable finite groups with arbitrarily small Sylow-integral invariant

Let 𝐺 be a finite group and let 𝛾 ⁡ ( 𝐺 ) = ∫ 1 0 𝑆 ⁢ 𝑃 ⁡ ( 𝐺 , 𝑥 ) 𝑑 𝑥 = ∑ 𝑝 ∈ 𝜋 ⁡ ( 𝐺 ) 𝑛 𝑝 ⁡ ( 𝐺 ) 𝑠 𝑝 ⁡ ( 𝐺 ) + 1 be the Sylow-integral invariant associated with the Sylow polynomial of 𝐺 . We show that small values of 𝛾 ⁡ ( 𝐺 ) do not force supersolvability. More precisely, if 𝐻 is any finite group and 𝑁 𝑡 = ∏ 𝑝 ∈ 𝜋 ⁡ ( 𝐻 ) 𝐶 𝑝 𝑡 , then 𝛾 ⁢ ( 𝐻 × 𝑁 𝑡 ) → 0 as 𝑡 → ∞ . Consequently, for every ɛ > 0 there exists a finite solvable group 𝐺 which is not supersolvable and satisfies 𝛾 ⁡ ( 𝐺 ) < ɛ . We also compute 𝛾 ⁢ ( 𝐶 𝑘 × 𝐴 4 ) explicitly and record infinite subfamilies with 𝛾 ⁡ ( 𝐺 ) < 6 / 5 .