A correction to the 41-structure of the Monster, a construction of a new maximal subgroup \(L_2(41)\) and a new Moonshine phenomenon. (Q2840175)

In [Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. Oxford: Clarendon Press (1985; Zbl 0568.20001); Lond. Math. Soc. Lect. Note Ser. 249, 198-214 (1998; Zbl 0908.20008); and Proc. Lond. Math. Soc., III. Ser. 84, No. 3, 581-598 (2002; Zbl 1017.20009)], it was stated that \(L_2(41)\) is not a subgroup of the Fischer-Griess Monster \(\mathbb M\). However, it was recently pointed out to the authors by A. V. Zavarnitsine that an argument in these works is invalid. In the present paper, the authors correct the result, by explicit computer calculations. Their main result is that there is exactly one conjugacy class of subgroups \(L_2(41)\) in \(\mathbb M\), such subgroups being self-normalizing and maximal. The authors note that the remaining cases of simple groups which might possibly be normal in still unknown almost simple maximal subgroups in \(\mathbb M\) are: \(L_2(13)\), \(U_3(4)\), \(U_3(8)\) and \(Sz(8)\). The obtained result leads to a new Moonshine phenomenon.