Sporadic simple groups and quotient singularities (Q2854099)

The present paper is devoted to prove the following statement:NEWLINENEWLINELet \(G\cong 6.\mathrm{Suz}\) be the universal perfect central extension of the Suzuki simple group and \(U\) be a \(12\)-dimensional irreducible representation of \(G\). Then the quotient singularity \(U/G\) is weakly exceptional but not exceptional, in the sense of [\textit{V. V. Shokurov}, J. Math. Sci., New York 102, No. 2, 3876--3932 (2000; Zbl 1177.14078)] and [\textit{Yu. G. Prokhorov}, Blow-ups of canonical singularities. A. G. Kurosh, Moscow, Russia, May 25-30, 1998. Berlin: Walter de Gruyter. 301--317 (2000; Zbl 1003.14005)].NEWLINENEWLINEAs consequence of this theorem, the authors get the following classification result: let \(G\) be a sporadic group or a central extension of one with centre contained in the commutator subgroup and let \(G\hookrightarrow \mathrm{GL}(U)\) be a faithful finite-dimensional complex representation of \(G\). Then the singularity \(U/G\) is: \begin{itemize}\item[-] exceptional if and only if \(G\cong 2.\mathrm{J}_2\) is a central extension of the Hall-Janko sporadic simple group and \(U\) is a \(6\)-dimensional irreducible representation of \(G\); \item[-] weakly exceptional but not exceptional if and only if \(G\cong 6.\mathrm{Suz}\) and \(U\) is a \(12\)-dimensional irreducible representation of \(G\).NEWLINENEWLINEThis result shows that among the sporadic simple groups, the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) are somehow distinguished from a geometric point of view. This motivates the author to pose the following question: ``Is there a group-theoretic property that distinguishes the groups \(\mathrm{J}_2\) and \(\mathrm{Suz}\) among the sporadic simple groups?''\end{itemize}