The maximal subgroups of the exceptional groups \(F_4(q)\), \(E_6(q)\) and \(^2\!E_6(q)\) and related almost simple groups.(English)Zbl 1523.20051
In this seminal paper, the author exhibits a complete list of all maximal subgroups of the finite simple groups of type \(F_{4}\), \(E_{6}\) and \(^{2}\!E_{6}\) over all finite fields. Prior to this paper, a complete answer was only known for \(F_{4}(2)\) [S. P. Norton andR. A. Wilson Commun. Algebra 17, No. 11, 2809–2824 (1989;Zbl 0692.20010)] and \(E_{6}(2)\) [P. B. Kleidman andR. A. Wilson Proc. Lond. Math. Soc., III. Ser. 60, No. 2, 266–294 (1990;Zbl 0715.20008)].
The main results are summarized in several tables (where the “subfield groups” are omitted) to which the reviewer refers the interested reader.
Let \(q\) be a power of the prime \(p\). The main novelty are the following.
In Table 2, two new maximal subgroups of \(E_{6}(q)\) appear:\[\mathrm{PSL}_{2}(13) \leq E_{6}(p) \quad\text{for } p \equiv 1,2,4 \mod 7, \quad p \equiv \pm 1,\pm 3, \pm 4 \mod 13\]and \(\mathrm{PSL}_{2}(19)\) in \(E_{6}(4)\).
In Table 3, two new maximal subgroups of \(^{2}\!E_{6}(q)\) appear, namely \(\Omega_{7}(3)\) is maximal in \(^{2}\! E_{6}(2)\) and \(\mathrm{PSL}_{2}(13)\) is maximal in \(^{2}\!E_{6}(p)\) for \(p \equiv 2,5,6 \mod 7\), \(p \equiv \pm 1,\pm 3, \pm 4 \mod 13\).
In the process of determining the maximal subgroups of \(F_{4}(q)\) for \(q\) even, a new maximal subgroup of the large Ree groups was found by the author. More precisely, there are exactly three conjugacy classes of maximal subgroups isomorphic to \(\mathrm{PGL}_{2}(13)\) in \(^{2}\!F_{4}(8)\), permuted transitively by the field automorphism (this subgroup was omitted in [G. Malle, J. Algebra 139, No. 1, 52–69 (1991;Zbl 0725.20014)]).
This beautiful paper is a valuable source of new information about the structure of simple groups of exceptional type (see also the author’s book [Maximal \(\mathrm{PSL}_2\) Subgroups of exceptional groups of Lie type. Providence, RI: American Mathematical Society (AMS) (2022;Zbl 1525.20014)]). As mentioned by the author, the techniques used in this paper are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for \(E_{6}\).
The main results are summarized in several tables (where the “subfield groups” are omitted) to which the reviewer refers the interested reader.
Let \(q\) be a power of the prime \(p\). The main novelty are the following.
In Table 2, two new maximal subgroups of \(E_{6}(q)\) appear:\[\mathrm{PSL}_{2}(13) \leq E_{6}(p) \quad\text{for } p \equiv 1,2,4 \mod 7, \quad p \equiv \pm 1,\pm 3, \pm 4 \mod 13\]and \(\mathrm{PSL}_{2}(19)\) in \(E_{6}(4)\).
In Table 3, two new maximal subgroups of \(^{2}\!E_{6}(q)\) appear, namely \(\Omega_{7}(3)\) is maximal in \(^{2}\! E_{6}(2)\) and \(\mathrm{PSL}_{2}(13)\) is maximal in \(^{2}\!E_{6}(p)\) for \(p \equiv 2,5,6 \mod 7\), \(p \equiv \pm 1,\pm 3, \pm 4 \mod 13\).
In the process of determining the maximal subgroups of \(F_{4}(q)\) for \(q\) even, a new maximal subgroup of the large Ree groups was found by the author. More precisely, there are exactly three conjugacy classes of maximal subgroups isomorphic to \(\mathrm{PGL}_{2}(13)\) in \(^{2}\!F_{4}(8)\), permuted transitively by the field automorphism (this subgroup was omitted in [G. Malle, J. Algebra 139, No. 1, 52–69 (1991;Zbl 0725.20014)]).
This beautiful paper is a valuable source of new information about the structure of simple groups of exceptional type (see also the author’s book [Maximal \(\mathrm{PSL}_2\) Subgroups of exceptional groups of Lie type. Providence, RI: American Mathematical Society (AMS) (2022;Zbl 1525.20014)]). As mentioned by the author, the techniques used in this paper are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for \(E_{6}\).
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20E28 | Maximal subgroups |
| 20D06 | Simple groups: alternating groups and groups of Lie type |
| 20G41 | Exceptional groups |
| 20B15 | Primitive groups |
Cite
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