On construction of certain Fischer embedded subgroups generated by 3-transpositions in \(F_{i_{22}}\) (Q6650413)

The sporadic group \(Fi_{22}\) of order \(2^{17}\cdot 3^{9} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13\) is described by \textit{B. Fischer} in [Invent. Math. 13, 232--246 (1971; Zbl 0232.20040)]. This group is generated by a conjugacy class \(D\) of 3-transpositions, that is involutions, any non-commuting pair of which has product of order 3. If \(G\) is generated by a class \(D\) of \(3\)-transpositions, then \(\mathrm{rank}(G)\) is defined as \(\max \big \{ |X| \; \big | \; X \subseteq D, \; [X, X]=1 \big \}\).\N\NIn the paper under review, the author obtains the following results. (1) He determines the rank of the groups \(F = 3\cdot Fi_{22}\), the Weyl group \(W\) of type \(E_{6}\) over fields of characteristic \(2\), the group \(M\) of shape \(3^{6} \rtimes U_{4}(2) \cdot 2\) and the group \(H_{1}= 3\Omega_{7}(3)\). (2) He proves that the subgroups \(\Omega^{+}_{8}(2)\cdot S_{3}\), \(3^{5}\cdot S_{6}\), \(3 \cdot U_{4}(3) \cdot 2\) and \(SU_{6}(2)\) are Fischer embedded and he gives an explicit construction for them.\N\NThe paper contains several misprints.