15.53 (2002)

*The set $S$ is finite and consists of 2 and 3. This follows from the complete list of finite simple groups $G$ with a module $V$ in characteristic $p$ satisfying those properties, which was determined in a number of papers, the last of which is (M. A. Grechkoseeva, S. V. Skresanov, Sibirsk. Elektron. Matem. Izv., 17 (2020), 585–589). Then the group $G$ must be either $3D_4(2)$ with $p = 2$ (V. D. Mazurov, Algebra Logic, 52, no. 5 (2013), 400–403), or $U_5(2)$ with $p = 3$ (A. V. Zavarnitsine, Siberian Math. J., 49, no. 2 (2008), 246–256, and M. A. Grechkoseeva, J. Algebra, 339, no. 1 (2011), 304–319), or $U_3(q)$, where $q$ is a special Mersenne prime, with $p = 2$ (A. V. Zavarnitsine, Algebra Logic, 45, no. 2 (2006), 106–116).