Definition of "finite group of Lie type"?

Question

The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from different Lie types). Call this collection of isomorphism classes of finite groups $\mathcal{S}$. But it's not clear to me that there is a similar consensus about the meaning of "finite group of Lie type". Probably most people would place the finite general linear groups on this list. Maybe also the Weyl group of the $E_8$ root system, which is related in the Atlas of Finite Groups to the simple group denoted $G=\mathrm{O}_8^+(2)$ as the group $2.G.2$.

The term "finite group of Lie type" comes up frequently in the literature (almost 500 times in a MathSciNet search of titles and reviews). There are two main directions in which such groups are approached, which might possibly lead to different lists. Often the starting point is a simple algebraic group over a finite field (with no nontrivial proper closed connected normal subgroups), though $\mathrm{GL}_n$ doesn't quite fit here.

(1) Steinberg's efficient organization in terms of finite fixed point groups under endomorphisms, denoted $G_\sigma$, is now often expressed in terms of "Frobenius morphisms" and their fixed points $G^F$. Basically the groups of interest then come in three flavors: split (Chevalley) groups, quasi-split (Hertzig, Steinberg, Tits) groups in types $A_n (n \geq 2), D_n, E_6$, or Suzuki/Ree groups in types $B_2, G_2, F_4$ with $p = 2, 3, 2$ as defining characteristic.

This leads to a collection $\mathcal{L}_1 \supset \mathcal{S}$ of (isomorphism classes of) Lie-type groups if one is allowed to take derived groups and to factor groups by subgroups of their centers. For this I certainly want to include general linear and unitary groups, so for type $A_{n-1}$ the collection $\mathcal{L}_1$ includes $\mathrm{GL}_n, \mathrm{SL}_n, \mathrm{PGL}_n, \mathrm{PSL}_n$ and related unitary groups. Often people specify at the outset a connected reductive group $G$ over a finite field, but it's best to assume that $G$ has a simple derived group for the current purpose.

(2) The Atlas focuses instead on the finite simple groups and then builds character tables for various related groups, as in my Weyl group example above. Here the simple group $G$ can be enlarged to a group in which $G$ is normal and the quotient is abelian, or can acquire nontrivial central extensions using its Schur multiplier. Starting with the collection $\mathcal{S}$, one obtains in this way a larger collection $\mathcal{L}_2$ of (isomorphism classes of) "groups of Lie type". Partly because of questions about general linear groups and the like, I'm left with a question:

How close to being the same are the collections $\mathcal{L}_1$ and $\mathcal{L}_2$? (Is there a natural way to define them to ensure equality?)

Answer 1

I feel a bit funny posting this as an answer to someone who has written a book with "finite groups of Lie type" in the title.

But I also find the matter both confusing and interesting (and full of conflicting terminology!), so here's my outside take on this, being a topologist of training:

(My own literature references include the standard sources: Carter, Gorenstein-Lyons-Solomon3..., as well as the recent (recommendable!) book by Malle-Testerman):

I thought the most general definition of a finite group of Lie type was close to your definition $\cal L_1$:

Definition ($\cal L_3$): A finite group of Lie type $G$ is a finite group obtained as ${\mathbf G}^F$, where $\mathbf G$ is a connected reductive algebraic group over an algebraically closed field of characteristic $p$, and $F$ is a Steinberg endomorphism (aka "twisted Frobenius", a composite of a Frobenius map and an automorphism of $\mathbf G$), or a group obtained from such a group ${\mathbf G}^F$ by modding out by a central subgroup, or passing to a normal subgroup with abelian quotient group.

Groups of this form will have many "Lie-like" properties, and in particular admit a canonical BN-pair structure. (Admitting a BN-pair structure would be another possible definition!)

Some authors restrict $\cal L_3$ by assuming $\mathbf G$ is semi-simple, or even simple, and some authors don't allow central quotients or passing to subgroups contained in the commutator subgroup. (One reason to allow the central quotients and subgroups is that otherwise the construction rarely produces simple groups: $|{\mathbf G}^F|$ only depends on the isogeny class of $\mathbf G$, e.g., adjoint and simply connected form have the same order, so ${\mathbf G}^F$ itself will never be simple unless $\mathbf G$ has associated Cartan matrix of determinant $1$.) Sticking with reductive (rather than simple or semi-simple) has the advantage that Levi subgroups in finite groups of Lie type are still finite groups of Lie type, which is useful if you're doing induction.

If you (for some reason) want to avoid tori $\mathbb F_q^{\times}$ and twisted products of non-trivial groups in this definition, but keep $GL_n(\mathbb F_q)$, you can maybe add the assumption to $\mathcal L_3$ that the root datum associated to $\mathbf G$ is indecomposable, and the associated root system not a product of two non-trivial root systems, or something like that? To an outsider this seems artificial?

However, not all classical groups will be in $\mathcal L_3$, e.g., $SO_{2n}^+(\mathbb F_q)$ (with the standard (classical!) definition). The problem arises from that its not connected as an algebraic group when $q$ is a power of two, so one has to pass to an index 2 subgroup, the kernel of the "pseudo/quasi-determinant", instead of the old-fashioned determinant. In particular the Weyl group of $E_8$, which is a central extension $2\cdot SO_8^+(8)$, is also not in $\mathcal L_3$, as Derek alluded to. ($W_{E_8}$ is "even further" freakish than $SO_8^+(2)$ since the central extension is related to that the index two subgroup $Spin^+_8(2)$ has non-generic Schur multiplier $\mathbb Z/2 \times \mathbb Z/2$, not zero as usual.)

I would (perhaps naively) argue that these groups also should not be considered finite groups of Lie type since they (unless I'm mistaken) do not admit the structure of a BN-pair; the structure of their p-radical subgroups is more chaotic (the Borel-Tits theorem fails).

Interpreting your $\mathcal L_2$ to consist of groups $G$ which has a filtration by normal subgroups $1< Z < H < G$, where Z is central in $G$, $G/H$ is abelian, and $H/Z$ is a simple non-abelian subquotient of a group in $\mathcal L_3$ (or equivalently $\mathcal L_1$), we will have $\mathcal L_3 \subseteq \mathcal L_2$.

With these interpretations, we can give the following response to your question:

$\mathcal L_1 \subseteq \mathcal L_3 \subseteq \mathcal L_2$, with $W_{E_8}$ or $SO_{2n}^+(\mathbb F_q)$, $q$ even, being examples of groups in $\mathcal L_2$ not in $\mathcal L_3$.

Note that a group like $GL_n(\mathbb F_p)$ of course would be in $\mathcal L_3$. Whether it is in $\mathcal L_1$ depends a bit on which $\mathbf G$ you are willing to start with.

Now, you may want to limit the class in $\mathcal L_2$ in some way to ensure that it is not larger than $\mathcal L_3$ and avoid the "pathologies". I would be surprised if there was a good way to do this, without referring back to $\mathcal L_3$: The main problem, I think, it that there is no really good "purely group theoretic" way of e.g., specifying that you only allow extensions "on top" by diagonal automorphisms (not field or graph), without getting back into the Lie theory. This is at least what goes wrong in the $SO_{2n}^+(\mathbb F_q)$, $q$ even, case. (Derek also hinted at this.)

You may also, conversely, argue that $\mathcal L_2$ is anyway too small, since it is not closed under iteratively putting things "on top" and "on the bottom"; to be "almost" of Lie type should be a bigger class... The cleanest fix for this is to go all the way and study the groups which are virtually a finite group of Lie type, i.e., has a finite index subgroup which a finite group of Lie type. Now, this class of groups is however of course just the class of virtually trivial groups, aka all finite groups...

This is not as silly as it sounds, since many deep open problems in finite group theory, such as Alperin's conjecture and its cousins, stem from the philosophy that all finite groups behave almost like finite groups of Lie type....

Answer 2

I (with co-authors) recently experienced a similar problem with the term "(finite) classical group". We initially tried to write down a precise definition, but eventually gave this up as a bad job, because we couldn't come up with anything that seemed right.

It is very unclear to me what you want to include in $\mathcal{L}_1$. Will you allow extensions of the simple group by field or graph automorphisms? You would certainly expect to find them in $\mathcal{L}_2$. Would you want to include the conformal classical groups, i.e. the normalizers in ${\rm GL}_n(q)$ of the symplectic, orthogonal and unitary groups?

For downward central extensions, it would seem natural to include extensions by the $p'$-part of the Schur multiplier in characteristic $p$ in $\mathcal{L}_1$. Isn't this what they call the simply connected version of the group? So you would get the spin groups (central extensions of orthogonal groups by a 4-group), which do not arise as classical groups. But interestingly, you would not get $2.O^+_8(2)$.

The collection $\mathcal{L}_2$ seems comparatively well-defined, but even there things become obscure when you try and extend upwards and downwards simutaneously. For simple groups $G$, even when $2.G$ and $G.2$ are specified, there will generally be either 0 or 2 isomorphism types of corresponding groups $2.G.2$, and perhaps just one of these will be a subgroup of a group in $\mathcal{L}_1$.