On cyclically symmetric presentations for \(SL_ 3\) and \(G_ 2\) (Q1312353)

Steinberg has found a presentation of \(\text{SL}(3,\mathbb{Z}/n\mathbb{Z})\) with generators \(y_ i\), \(i \in \mathbb{Z}/6\mathbb{Z}\), and relations invariant under the cyclic symmetry \(y_ i \rightarrow y_{i + 1}\). In the paper, a more general group \(L(m,n)\) is defined by a similar presentation with generators \(y_ i\), \(i \in \mathbb{Z}/m\mathbb{Z}\) (so \(L(6,n)\) is isomorphic to \(\text{SL}(3,\mathbb{Z}/n\mathbb{Z})\)). It is conjectured that \(L(7,p)\) is isomorphic to \(G_ 2(\mathbb{Z}/p\mathbb{Z})\) for \(p = 0\) and all odd primes \(p\). The conjecture is proved for \(p = 3\) and 5, and other results in its support are obtained.