For a finite group \(G\) , let \(\nu _p(G)\) be the number of Sylow \(p\) -subgroups of \(G\) , let \(\sigma _p(G)\) be their common order, and \( \gamma (G)=\int _0^1\!\!\sum _{p\in \pi (G)}\!\!\nu _p(G)\hspace{1.111pt}x^{\sigma _p(G)}\,dx \,=\!\!\sum _{p\in \pi (G)}\frac{\nu _p(G)}{\sigma _p(G)+1}\hspace{0.55542pt}. \) A conjecture attributed to Anabanti and Asboei asserts that, for every finite nonsolvable group \(G\) , the equality \(\gamma (G)=9/2\) holds if and only if \(G\cong A_5\) . We disprove this assertion by an explicit central extension of \(A_5\) . More generally, we prove an exact compensation formula for direct products \(A_5\hspace{0.55542pt}{\times }\hspace{1.111pt}N\) , where \(N\) is finite nilpotent. The formula reduces the equality \(\gamma (A_5\hspace{0.55542pt}{\times }\hspace{1.111pt}N)=9/2\) to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of \(N\) . Taking \( N=\mathsf C_2\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_7\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{11}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{13}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{17}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{19}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{29}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{71}\hspace{0.55542pt}{\times }\hspace{1.111pt}\mathsf C_{83}, \) the loss in the old \(2\) -Sylow contribution is exactly compensated by the new normal Sylow subgroups. Hence \(A_5\hspace{0.55542pt}{\times }\hspace{1.111pt}N\) is nonsolvable, is not isomorphic to \(A_5\) , has solvable radical \(N\) , and nevertheless satisfies \(\gamma (A_5\hspace{0.55542pt}{\times }\hspace{1.111pt}N)=9/2\) . Several further exact compensation certificates are also recorded.
