A NATURAL EXISTENCE PROOF FOR JANKO’S SPORADIC GROUP $\mathrm{J}_{1}$

ABSTRACT. Using the second author’s deterministic algorithm [8], which constructs all the finitely many simple groups $G $ having a 2-central invo-lution, say $t $ , such that $C_{G}(t) $ is isomorphic to a given group $H\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}^{\mathrm{r}}\mathrm{i}\mathrm{n}\mathrm{g}$ certain natural conditions, in this article we give a new and in some sense natural existence proof for Janko’s first sporadic simple group $\mathrm{J}_{1}[6] $. 1