The maximal subgroups of F4 , E6 and 2E6 and related almost simple groups, Errata

I distinguish four types of corrections, in order of increasing seriousness:

• (Extra) Additional information that was not available at the time of writing, or that I did not know about.

• (Improve) Typographical issues, where what is written is still correct, but there is a nicer way of phrasing it, or I could choose a better symbol.

• (Typo) Typographical errors, where I have spelled a word wrongly, used the wrong symbol, and so on.

• (Error) Errors in proofs or statements.

When I give each correction, I will label it with one of these monikers.

• (i) (Error) In Table 8 there is a torus normalizer with structure \((q^2+1)^2\cdot (4 \circ \mathrm {GL}_2(3))\). The group \(4 \circ \mathrm {GL}_2(3)\) is not a subgroup of the Weyl group of type \(F_4\), and the correct group should be \(\mathrm {SL}_2(3)\rtimes 4\), which is the centralizer of an element of order \(4\) in \(W(F_4)\). The original paper of Liebeck–Saxl–Seitz also contains this error, and I copied their tables over without double-checking.

  Thanks to Mikko Korhonen for noticing this.

• (ii) (Error/Typo) In Table 8 the large Ree subgroups have a badly explained structure. I have written \({}^2\!F_4(q_0)\) in analogy with the fact that \(\mathrm {SU}_n(q)\leq \mathrm {SL}_n(q^2)\). This doesn’t really work for large Ree groups, and the more standard notation is that \({}^2\!F_4(2)\leq F_4(2)\), for example. So this line should read that \({}^2\!F_4(q)\leq F_4(q)\).

  Thanks to Tim Burness for mentioning this.

  More importantly, when writing this erratum I noticed that the condition that \(q\) is an odd power of \(2\) is missing from this, but since there is no such group as \({}^2\!F_4(4)\) this is heavily implied, even if it is an error that it is not present.

• (iii) (Error) In Table 9 the subgroup \(d^2.(\mathrm {P}\Omega _8^+(q)\times ((q-1)/d)^2/e).d^2.\Sym (3)\) is correct. Comparing this with Table 10, we find the correct version should be \(d^2.(\mathrm {P}\Omega _8^+(q)\times ((q + 1)/d)^2/e').d^2.\Sym (3)\), not the \(d^2.(\mathrm {P}\Omega _8^+(q)\times ((q + 1)/d)^2).d^2.\Sym (3)\) as given.

  Thanks to Melissa Lee for noticing this.

• (iv) (Error) In Table 9 the novelty subgroup should be \([q^{31}].(\SL _2(q) \times \SL _2(q) \times SL_3(q)).(q-1)^2/e\). There’s a missing \((q-1)\) factor in the torus (obvious) and the action when removing the centre is to quotient by a diagonal \(e\) in the \(\SL _3(q) \times (q-1)^2\) subgroup. This yields a central product, so we can write the structure above.

  Thanks to Tim Burness for highlighting this.