Constructing the Monster amalgam. (Q2498860)

The Monster group \(M\) in generated by two subgroups \(C\) and \(N\), \(C\) the centralizer of a 2-central involution and \(N\) the normalizer of a fours group consisting of 2-central involutions. These groups play an important role in constructing and proving uniqueness of the Monster group. In this paper, the author considers a group \(C\) such that \(O_2(C)\) is extraspecial of order \(2^{25}\) and \(C/O_2(C)\cong Co_1\). Further \(C/O_2(C)\) acts on \(O_2(C)/Z(O_2(C))\) as \(Co_1\) on \(\Lambda/2\Lambda=\overline\Lambda\), \(\Lambda\) the Leech lattice. It is well known that there are two such groups \(C\). Let now \(t\in O_2(C)\) be an involution such that \(tZ(O_2(C))\in\overline\Lambda_4\). Set \(U=C_C(tZ(O_2(C)))\). Suppose there is a group \(N\) such that \(U\leq N\) and \(|N:U|=3\), then \(C\) is uniquely determined. The author then shows that for \(U_0=C_U(\langle t,Z(O_2(C))\rangle)\), \(\text{out}(U_0)\cong\Sigma_3\).