On the simplicity of \(\cdot 2\) and \(\cdot 3\) (Q1183999)

The Conway group \(\cdot 0\) [\textit{J. H. Conway}, Finite simple groups, 215-247 (1971; Zbl 0221.20014)] is the automorphism group of the Leech lattice \(L\) in 24-dimensional Euclidean space. The groups \(\cdot 2\) and \(\cdot 3\) are defined as stabilizers \(\cdot 0_ a\) in \(\cdot 0\) of lattice vectors \(a\) with scalar products \((a,a)=2\cdot 16\) resp. \(3\cdot 16\). For \(v\in L\) set \(L_ i(v,m)=\{x\in L\mid (x,x)=i\cdot 16\), \((x,v)=m\}\). The author proves the simplicity of \(\cdot 2\) and \(\cdot 3\) by showing that \(\cdot 0_ a\) acts primitively on \(L_ 2(a,8)\) resp. \(L_ 3(a,32)\) with simple stabilizer \(\cdot 0_{a,b}\) and without a regular normal subgroup, where the simplicity of \(\cdot 0_{a,b}\) in turn follows from its primitivity on \(L_ 2(9,8)\cap L_ 2(b,16)\) resp. \(L_ 2(a,24)\cap L_ 2(b,24)\) with stabilizer \(\cdot 0_{a,b,x}\cong M_{22}\). \textit{M. Suzuki} [Group Theory, Proc. Conf. Singapore 1987, 195-206 (1989; Zbl 0662.20009)] offers a short elementary proof of the simplicity of \(\cdot 2\) and \(\cdot 3\) which avoids these orbit calculations and mainly rests on the structure of the normalizer in \(\cdot 0\) of a subgroup of order 23.