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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.20976 |
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| _version_ | 1866911725272432640 |
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| author | Zhang, Yutong Yang, Yaoran |
| author_facet | Zhang, Yutong Yang, Yaoran |
| contents | Let \(ν_p(G)\) be the number of Sylow \(p\)-subgroups of a finite group \(G\), let \(σ_p(G)\) be their common order, and set \[
γ(G)=\int_0^1\sum_{p\inπ(G)}ν_p(G)x^{σ_p(G)}\,dx
=\sum_{p\inπ(G)}\frac{ν_p(G)}{σ_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \(γ(G)=9/2\) 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 \(γ(A_5\times N)=9/2\) 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 \(γ(G)=9/2\). Several further explicit compensation certificates are also recorded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture Zhang, Yutong Yang, Yaoran Group Theory Let \(ν_p(G)\) be the number of Sylow \(p\)-subgroups of a finite group \(G\), let \(σ_p(G)\) be their common order, and set \[ γ(G)=\int_0^1\sum_{p\inπ(G)}ν_p(G)x^{σ_p(G)}\,dx =\sum_{p\inπ(G)}\frac{ν_p(G)}{σ_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \(γ(G)=9/2\) 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 \(γ(A_5\times N)=9/2\) 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 \(γ(G)=9/2\). Several further explicit compensation certificates are also recorded. |
| title | A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.20976 |