7.17 (1980)

Editors’ remarks: This is proved for $G$ solvable (G. E. Wall, J. Austral. Math. Soc., 2 (1961–62), 35–59) and for symmetric groups $S_n$ for sufficiently large $n$ (M. Liebeck, A. Shalev, J. Combin. Theory, Ser. A, 75 (1996), 341–352); it is also proved (M. W. Liebeck, L. Pyber, A. Shalev, J. Algebra, 317 (2007), 184–197) that any finite group $G$ has at most $2^{C |G|^{3/2}}$ maximal subgroups, where $C$ is an absolute constant.

No, not always (R. Guralnick, F. Lübeck, L. Scott, T. Sprowl); see (C. P. Bendel, B. D. Boe, C. M. Drupieski, D. K. Nakano, B. J. Parshall, C. Pillen, C. B. Wright, in: Developments and retrospectives in Lie theory. Algebraic methods. Retrospective selected papers based on the presentations at the seminar “Lie groups, Lie algebras and their representations”, 1991–2014, Springer, Cham, 2014, 51–69). Infinitely many counterexamples were found in (F. Lübeck, Trans. Amer. Math. Soc., 373, no. 4 (2020), 2331–2347).