Some sporadic groups as Galois groups (Q1114767)

In this paper it is proved that the sporadic simple groups \(He\), \(Fi_{22}\), \(J_4\), have GAR-realizations over \(\mathbb{Q}(T)\). According to Matzat's definition [\textit{B. H. Matzat}, Invent. Math. 80, 365-374 (1985; Zbl 0567.12015)], these groups are Galois groups of regular extensions of \(\mathbb{Q}(T)\) with two additional properties. As a consequence of this result and of previous work of Matzat and others, it is obtained that all sporadic simple groups have GAR-realizations over \(\mathbb{Q}^{ab}(T)\).