Uniform and natural existence proofs for Janko's sporadic groups \(J_2\) and \(J_3\) (Q699732)

\textit{Z. Janko} [Symp. Math. 1, 25-64 (1969; Zbl 0182.35304)] gave evidence for the simple sporadic Janko groups \(J_2\) and \(J_3\) from their common involution centralizer \(H\cong 2^{1+4}:A_5\). The existence and uniqueness of \(J_2\) and \(J_3\) was originally proved by \textit{M. Hall} jun. and \textit{D. Wales} [J. Algebra 9, 417-450 (1968; Zbl 0172.03103)] and \textit{G. Higman} and \textit{J. McKay} [Bull. Lond. Math. Soc. 1, 89-94 (1969; Zbl 0175.30103)], respectively. In this paper, the author gives a uniform and natural approach to the existence proofs. It is uniform, because it uses a quite general deterministic algorithm described by G. O. Michler in an unpublished paper. It is natural, because it starts from a canonical permutation representation \(\rho(A_5)\). Then, based on \(\rho(A_5)\), a finite presentation for \(H\) is constructed. Finally, as result of Michler's algorithm, the embeddings of \(J_2\) in \(\text{GL}_{14}(11)\) and \(\widehat{J_3}\) in \(\text{GL}_{18}(31)\) are obtained. All computations can easily be verified by means of MAGMA.