# Choice of bilinear forms for simple groups

In the finite simple groups, some groups are obtained from (non-degenerate) bilinear forms as follows:

Let $$V$$ be a vector space over a finite field $$\mathbb{F}_q$$. Let $$(\cdot,\cdot)$$ be a non-degenerate bilinear form. Consider the group

$$G=\{ T\in {\rm End }(V) : (Tv,Tw)=(v,w)\mbox{ for all } v,w\in V\}.$$ Since form is non-degenerate, $$G$$ is subgroup of $$\mathrm{GL}(V)$$. Next, since vector space is finite, this group is finite.

By Jordan-Hölder, the group $$G$$ gives rise to some simple groups in its composition series.

Among finite simple groups, there are some choice on bilinear forms $$(\cdot, \cdot)$$ such as symmetric, alternating, sesquilinear.

But for (non-degenerate) forms other than these types, we get a finite group $$G$$ preserving the form, and by Jordan-Hölder, it gives simple groups in its composition series. Then, why other forms are not considered generally?

Turning this question in a different way, we may raise following concrete specific question:

Is there a non-degenerate bilinear form on a vector space $$V$$ over finite field $$\mathbb{F}_{q}$$ such that the full subgroup of $$\mathrm{GL}(V)$$ preserving this bilinear form is nilpotent group or solvable group for all $$q$$, except finitely many.

(In any introductory text/notes on simple groups, it is directly stated — consider symmetric/alternating/sesquilinear form on vector space and do this....; without stating why other types of forms can be ignored.)

I think the key word you need here is reflexive. A bilinear form $$\beta$$ is reflexive if $$\beta(x,y)=0 \Longrightarrow \beta(y,x)=0.$$

It's pretty clear that any sensible notion of orthogonality (and, thereby, a sensible geometry on which a group can act) is going to require a reflexive bilinear form. If you are willing to go along with this, then the answer to your question is No, there are no other forms. More precisely, the following theorem answers the question.

Theorem Let $$V$$ be a vector space with $$\dim(V)\geq 3$$ and $$\beta$$ a reflexive non-degenerate $$\sigma$$-sesquilinear form. Then $$\beta$$ is alternating, symmetric or a scalar multiple of a $$\sigma$$-Hermitian form.

You can find a proof in these notes of mine or in Peter Cameron's notes.

Caveat. Re-reading this theorem, I can't remember exactly where the lower bound on dimension is used in the proof. I guess it's possible that in dimension $$1$$ or $$2$$, there could be interesting geometries associated with other types of reflexive form.

• I meant to say, by the way, that I had exactly the same question in my head when I first studied bilinear forms many moons ago. It does all seem a bit random when you first see it.... Jan 23 '17 at 12:06
• I think it's called reflexive rather than reflexible. (Both your own course notes and Peter's notes seem to confirm this.) I've never encountered "reflexible" before, in any case... Jan 23 '17 at 12:41
• @TomDeMedts, of course it is! Sorry, was typing without thinking. Shall correct now. Jan 23 '17 at 12:55
• A few 'reflexible's survive. In dimension 1, every $\sigma$-sesquilinear form is trivially a scalar multiple of a $\sigma$-Hermitian form, so the interesting geometry, if any, is in dimension 2. Dec 14 '20 at 22:27
• @LSpice, oops,fixed now. Dec 16 '20 at 14:32

Books on finite simple groups generally do not assume the reader has expertise in the theory of linear algebraic groups, and so to "define" the objects of interest they have to resort to making a long list of constructions that may seem unmotivated if one doesn't have experience with Lie theory over $\mathbf{C}$. But if one does assume familiarity with the Borel-Tits structure theory over arbitrary fields then one can give a uniform and conceptual definition for nearly all of these groups: $G(k)/Z_G(k)$ for a finite field $k$, a connected semisimple $k$-group $G$ that is absolutely simple and simply connected, and its scheme-theoretic center $Z_G$. Of course, in practice one wants and needs a more tangible description than that (such as to account for the Ree and Suzuki groups, to handle special behavior over $k$ of size 2 or 3, and to get experience with examples), but that succinct definition turns out to be enough to prove quite a lot without ever needing to mention linear algebra over finite fields (e.g., a uniform proof that such groups are simple away from a very tiny list of exceptions)!

So you seem to have a mistaken impression about the role of those bilinear forms (even setting aside that to handle all positive characteristics in a uniform manner, which is to say to treat characteristic 2 on equal footing with the rest, one should use non-degenerate quadratic forms -- for appropriately defined notion of "non-degenerate" that works over all fields and rings -- rather than non-degenerate symmetric bilinear forms). The reason for considering such structures and guessing that they might even account for "nearly all" finite simple groups is not initially motivated by the idea that preserving such a form gives a finite group and one hopes that it is nearly simple (in the sense that passing to a Jordan-Holder series, or a very mild kind of composition series with the derived group and a central quotient thereof). Rather, it is due to the very specific way in which certain linear algebra constructions account for nearly the entirety of the classification of non-abelian simple Lie algebras and non-abelian simple Lie groups over $\mathbf{C}$ in a manner that is ultimately "defined over $\mathbf{Z}$". That suggests the possibility (originally due to Chevalley, building on much case-by-case work done by Dickson, Dieudonne, and others) to imitate such constructions over finite fields to get finite groups whose internal structure (e.g., simplicity) can be proved and studied in a unified manner.

The work of Borel and Tits (building on Chevalley's work over algebraically closed fields) provides a completely uniform structure theory for connected semisimple algebraic groups $G$ over arbitrary fields $k$ (not just finite fields), involving a classification expressed in terms of root systems and Galois cohomology (and making no mention whatsoever of linear algebra constructions or bilinear/quadratic forms). This gives rise to a general procedure for describing all possibilities for $G$ that are absolutely simple (in an algebraic group sense) in terms of a very specific array of algebraic structures: symplectic forms, non-degenerate hermitian forms, non-degenerate quadratic forms, Jordan algebras, octonion algebras, etc. The link to root systems also provides a systematic way to prove simplicity results for $G(k)$ or groups very near to $G(k)$ (e.g., quotient by center when $G$ is simply connected and $k$ isn't too small in a universal sense)

In the special case of finite fields (whose Galois cohomology is particularly simple, due to a combination of a result of Lang on torsors over finite fields and the pro-cyclicity of their absolute Galois group with "Frobenius" generator) some of these auxiliary algebraic structures only exist in a very limited manner (e.g., no non-trivial finite-dimensional central division algebras over such fields). Consequently, up to a very tiny list of exceptions, the exhaustive list of such groups $G(k)$ turns out to be explained by a very limited set of linear algebra constructions related to non-degenerate hermitian (= sesquilinear) forms, non-degenerate quadratic forms, and symplectic forms. And from knowledge of exceptional isogenies among such $G$'s one is led to the additional Ree and Suzuki groups in characteristics 2 and 3 by a very conceptual process (not how they were first discovered, but that's a separate matter).

So the reason one focuses largely on those specific constructions when studying "finite groups of Lie type" is a combination of (i) experience with Lie theory over $\mathbf{C}$, (ii) the fact that in the classification of root systems all but 5 are explained by specific linear algebra constructions, and (iii) the very simple nature of Galois theory for finite fields. (That the finite simple groups obtained in this way happen to account for nearly all finite simple groups is a real miracle.)