A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture

Abstract

Let \(\nu_p(G)\) be the number of Sylow \(p\)-subgroups of a finite group \(G\), let \(\sigma_p(G)\) be their common order, and set \[ \gamma(G)=\int_0^1\sum_{p\in\pi(G)}\nu_p(G)x^{\sigma_p(G)}\,dx =\sum_{p\in\pi(G)}\frac{\nu_p(G)}{\sigma_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \(\gamma(G)=9/2\) precisely when \(G\cong A_5\). We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for \(A_5\) with an arbitrary nilpotent factor. The formula reduces the equality \(\gamma(A_5\times N)=9/2\) to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of \(N\). Taking \(N=\C_2\times\C_7\times\C_{11}\times\C_{13}\times\C_{17}\times\C_{19}\times\C_{29}\times\C_{71}\times\C_{83}\), the loss in the \(2\)-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently \(G=A_5\times N\) is nonsolvable, is not isomorphic to \(A_5\), has solvable radical \(N\), and nevertheless satisfies \(\gamma(G)=9/2\). Several further explicit compensation certificates are also recorded.