Last updated: 14 January 2015
We will begin by recalling, for the purpose of comparison, the principal of the construction of the non-normal forms (i.e. not of type "Tôhoku") of the simple groups, by starting from the normal forms.
Let be a field of characteristic or a complex semisimple Lie algebra, a system of simple roots of the Dynkin diagram of (in the following we will assume that we can distinguish the roots of different lengths), the "Tôhoku group"  of type over and let and defined as in . To every automorphism of there corresponds an isometric permutation of which extends to an isometric permutation of the set of all roots, this will also be denoted by We will denote the automorphism of defined by For every root this automorphism transforms into By abuse of language, the automorphisms of which are the product of by an inner automorphism will be called birational.
To each endomorphism of on may associate an endomorphism of defined by Every endomorphism which is a product of a birational endomrphism with will be said to belong to
If is a galois extension of the group of a field an of may be defined as the group of elements fixed by an automorphism group of the image of by a monomorphism (the group of automorphisms of such that belongs??? to for every In particular, to each monomorphism one may make a corresponding of with the help of the monomorphism defined by The formes of and which have been considered by D. HERTZIG, R. STEINBERG  and the author  and , are of this type.
If one ignores the length of roots, three Dynkin diagrams acquire an automorphism of order 2; these are the diagrams of and To this automorphism, there corresponds a permutation of which preserves the angles between pairs of roots (but not their lengths), this extends to a permutation of the set of all roots which has the same property, and is also denoted In certain special cases, one may also associate an automorphism, or more generally an endomorphism of no longer "birational" but only "rational". More precisely, if or (resp. and (resp. 3), a simple calculation shows that there exists an endomorphism of defined by where or according to whether is a "long" or "short" root. It is clear that where is the Frobenius of Assume that possesses an endomorphism such that Then is an involutive automorphism of the group In particular, if is perfect, and are surjective, and is an automorphism of In all cases, we denote or the group of elements of such that (elements invariant by
Let (resp. be the subgroup of generated by the (resp. the and (resp. Then one has the following theorem:
If has more than or elements, the group generated by and which is non other than the derived group of is simple. Furthermore, (in particular, if is perfect, and where is a subgroup of which contains and
The part of this theorem which concerns and will be proved later, after another slightly different definition of these groups. For the case of we refer to .
The notations are as in paragraph 1, with the Weyl group of or ( defined as in , the group formed by the the group generated by and the the homomorphism with kernel defined by the group of elements of invariant by
It is clear that In fact, we have the following theorem:
The proof of this is done by explicitly constructing for each an in (the three cases, are treated separately). In the following we always assume that the are chosen in this fashion.
Given we put and, when Every element may be put, in one and only one fashion, in the form of a product with and (theorem 2 of ). On the other hand, and sends each of the groups and into itself (for and this is evident; for this is because is the normalizer of and is the centralizer of and of it follows that if belongs to then the same is true of and Thus,
Every element of (resp. may be put, in one and only one fashion, in the form of a product with (resp. and
We will also give, for each of the groups and some indications of the structure of and
Case (resp. is composed then of the identity and the operation which transforms each root to its opposite. exchanges and then by virtue of theorem 3. It follows that is doubly transitive on the space of left cosets of in In effect, one may canonically identify with the union of and a point the image of the identity element, and the subgroup of acts transitively on itself and preserves We will rediscover later (paragraphs 4, 5, 6) this doubly transitive representation of The subgroup is isomorphic to the multiplicative group of (resp. to a subgroup satisfying the conditions of theorem 1). is nilpotent of class 1 (resp. 3) and the quotients of its ascending central series are all isomorphic to the additive group (the precise structure of will be given in paragraph 6). If one denotes the simple roots by and with the short root, one has, with an appropriate normalization of the
Case is then a dihedral group of order 16. is isomorphic to Further there exist subgroups and respectively isomorphic to and to the "unipotent" of and having properties completely analogous to those ot the of the Tôhoku groups, so that, for example:
|and are invariant under|
|the inner automorphisms corresponding to elements of permute the and the in the same fashion that the euclidean automorphisms of figure 1 permute the half lines that form it (one sees that the "Dynkin diagram" of is a linear sextuple, which somewhat breaks the monotonie|
|If groups are chosen from the and corresponding to the half lines of figure 1 contained in an acute algebra ("convex cone") every element of the group which they generate can be put, in a unique fashion, in the form of a product with|
|if and are two acute angles,|
|is generated by the|
|for every and (resp. and generate a group isomorphic to (resp. to|
|if (resp. denotes the group generated by and (resp. the sets and constitute, with the usual incidence relation , a generalised octagon (cf. , Appendix);|
|one has also a "geometric interpretation" of as the group of automorphisms of this octagon;|
Let be the finite field with (resp. elements. We see easily that the Frobenius (resp. possesses a square root, d'aillers?????? unique, if and only if In this case, one may define the finite groups and (resp. which are simple when by virtue of Theorem 3 (in the case it is remarked that, because of the hypothesis made on is not a square in from which it results that and generate The orders are easily calculated from the results of paragraph 2, and one finds
One notes that is not divisible by a property that is not shared by any other known simple group. R. REE notes that one may show that the are also new simple groups by using the fact that is divisible by and not by Finally one may follow the method of Artin to give an analogous result for The are the groups of SUZUKI  (cf. number 4.3).
We will employ in many instances the following abbreviated notation: "A projective space "A projective space in which is given a homogeneous system of coordinates On the other hand, a variety will always be assimilated from the set of its rational points over
Let be a field of characteristic and a 3-dimensional projective space over The lines of may be specified in terms of their "Pücker coordinates" which satisfy the relation We consider the set ("linear complex") of lines such that These are represented, in a 4-dimensional projective space by the points of a hyperquadric with equation Now we assume that the characteristic of is equal to
The hyperplanes tangent to then meet at a point, with coordinates and this permits the definition (by "projection" from this point), a "rational" injection of into a projective space by
The Plücker coordinates of the lines of will be denoted To the set of lines of which contain a given point of correspond a line of thus a line of for which the Plücker coordinates are, after an elementary calculation, and which, in particular, belong to the set of lines of with equation In this way a sort of duality is established between and formed by two injections and such that if and denote, respectively, a point of and a line belonging to the properties of and are equivalent.
We will denote by (resp. the group, isomorphic to of Tôhoku, of projectivities of preserving (resp. of projectivities of preserving The "duality" between and "induces" a monomorphism defined by for each and each line
Now we assume that the Frobenius of possesses a square root and we consider the semilinear map defined by From number 4.1, it follows immediately that and are points of the relation is in the line is symmetric; thus functions as a sort of polarity. We will denote by the set of points such that the point is in its "polar"
We introduce now, in without the plane ("plane at infinity"), with the nonhomogeneous coordinates An elementary calculation shows that is composed of the point which we denote by and the set completely "at finite distance", of the equation
We see easily that there exists a monomorphism defined by for all and
We will denote by the group of elements such that (group of projectivities "convserving is not other than the group (the link between the two definitions is apparent). It is clear that is contained in the group of all projectivities of which preserve the set this will be denote by (in fact, we will see later). The group of elements of where the determinant is a 4th power, will be denoted
The projectivities are in For and this follows from an easy calculation; for the projectivity it is immediate that it is in Further, and giving that if and only if Finally, one verifies easily that and are groups.
If has more than two elements, the are the only elements of that preserve the points and
A projectitivy that preserves and preserves the plane which is in effect the only plane containing where the intersection with is reduced to this point. It follows that a projectivity that preserves and is a linear transformation represented by a linear and homogeneous substitution on Taking into account the fact that and are preserved, we see that if we make the substitution in the function this must be multiplication by a constant, but it is also the same as the "homogeneous components" because of the linear independence of multiplicative endomorphisms of (cf. , paragraph 7, no. 5) (we use here the fact that because one must assume that the endomorphisms and are all different). It follows that the projectivity in question is of the form and the proof is completed by a simple calculation.
The group (and a fortiori is doubly transitive on If has more than 2 elements, (resp. is generated by the projectivities (resp. and Every element of may be put, in one and only one fashion, in one of the two forms or Finally,
The group generated by the the and is doubly transitive on in effect, the subgroup which preserves is transitive on and "moves" The theorem is thus an immediate consequence of lemma 1 and the fact that is simply transitive on
It suffices to verify that and belong to
Remark. We will note that theorem 4 has been established solely on the basis of the analytic definition (4.2.1) of without making use of (if one takes into account the note
The equation (4.2.1) shows that every line passing through and not contained in the plane meets in one and only one point. Taking account of the "homogeneity" of (Theorem 4), one deduces immediately that
Every line in meets in 0, 1 or 2 points. The lines passing through a given point determine the set and those that do not meet any other (tangents) form a plane.
When is the finite field with elements, is formed by points.
We denote by the projective space of dimension over for some A set of points of this space is called an if every line meets it in 0, 1, or 2 points. The sets of this type have been abundantly studied by B. Segre and his school (cf. notably , , ). They have given, in particular, that
(Segre-Barlotti) If a in is a conic, a in is a quadric, and there does not exist a in for
In characteristic 2, the situation is more complicated. In there exist which are not conics (for example the set formed by the points of a conic and the meeting point of the tangents to it). The example of shows that there exist also, in which are not quadrics (the example given by B. Segre in  for is a special case of this one We mention also the following result (unpublished) of the author:
Let be a in Assume that there exists on a point such that the group of projectivities preserving and this point possesses an invariant subgroup which is simply transitive on and at least one cyclic subgroup of order Then or If is, in a system of affine coordinates appropriately chosen, a "curve" with equation completed by the "point at infinity on one of the coordinate axes", denotes any automorphism of having as its unique fixed point. If is a quadric, it is the set defined in section 4.2.
For in characteristic 3 one can give results completely analogous to those of numbers 4.1 to 4.3. We will limit ourselves to indicating the main lines. All that follows results by way of elementary calculations.
Let be a field of characteristic 3, a 6 dimensional projective space over the hyperquadric with equation the set of lines (all belonging to defined by the equations the group of projectivities of reserving (thus also (Tôhoku group over and the 5-dimensional variety representing in the 13-dimensional projective space
On and are polynomials in the and which also satisfy the relation This permits the definition of a "rational" injection ("projection" from the linear variety of in a hyperquadric of a 6-dimensional projective space We will denote by etc., the Plücker coordinates of the lines of If is a point of the set of lines of which pass through the point has image in (by the transfer of (5.2)) which is a line belonging to the set defined by the relations (5.1) where one replaces by and such that In this way we find two injections and establishing a sort of duality between and Then one may, exactly as in paragraph 4, define the group the monomorphism (where the existence results immediately from the fact, easily vareifiable, that the projectivities of induced by the elements of leave invariant and, if the Frobenius possesses a square root the "polarity" the monomorphism the set the groups and The point with coordinates belongs to and the set which is completely disjoint from the hyperplane has for equations, in non-homogeneous coordinates The following projectivities belong to (we leave to the reader the task of completing the definition expressing that (5.3) is preserved) and one has the
is doubly transitive on If has more than three elements, it is generate by the and Every element of may be put, in one and only one fashion, in one of the two forms or
|1.||The determinants of and are 7th powers, so is a subgroup of which is anyway already true (and well known) for|
|2.||There is no doubt that, just as and may be defined as the group of elements of (of Tôhoku) which "preserve" a "polarity" of the geometry associated to it (cf. , notably paragraph 11), and that the elements of the geometry incident to their "polar" form the generalised octagon in question in paragraph 2.|
Let be a doubly transitive group of permutations of a set such that the subgroup leaving fixed a point of possesses an invariant subgroup simply transitive on and two points of the group of elements of preserving and the conjugate of which has as a fixed point, the group generated by and the element of which moves to the element of which moves to the element of which moves to and the transformation which exchanges and
The group is manifestly transitive on it thus contains all the conjugates of in It is thus the smallest invariant subgroup of containing
being simply transitive on the set one may endow it with the structure of the "abstract" group (which will be denoted additively) having for the identity element and such that with (we put The elements of are automorphisms of the action of on will be denoted multiplicatively (i.e. we will put We easily see that (resp. is composed of transformations with (resp. The transformation belongs to and thus one may write Recalling the definition of and we see that and with Taking account of one then finds the relation Generally, we consider, for every the transformations and with these both move to and to and thus their quotient is an element of which one may write in particular We also note that putting in (6.6), it becomes
Let be the group generated by the the and It is clear that On the other had, we check easily that the transformations (6.2) and (6.3), with form a group which evidently contains and and thus It follows that
An easy computation shows that the preceding results admit a converse. More precisely,
Let be a group denoted additively, a group of automorphisms of and a permutation of We assume that there exists a pemutation a map and elements and such that one has the identities (6.4), (6.5), (6.6) and (6.7) for all and and let is the subgroup generated by the the and We also denote by the set obtained by adjoining a point to and we put and Then, the transformations of defined by (6.2) and (6.3), with (resp. form a doubly transitive group. The transformations (6.1) form an invariant subgroup of the group of elements of which preserve it is generated by any two conjugates of in is an invariant subgroup of and Finally, one may obtain in this way every doubly transitive group, such that the subgroup obtained by fixing a point possesses a simply transitive invariant subgroup on the set on which the group acts, minus the the point in question.
|1.||Let be a field, (additive group), (acting in the obvious way on and whence and Then and|
|2.||Let be a field of characteristic 2 where the Frobenius possesses a square root with the group law acting on in the following way: and, putting with defined by (4.2.1), whence and Then and are the group and of paragraph 4.|
|3.||Let be a field of characteristic 3 where the Frobenius possesses a square root with the group law acting on in the following way, and, putting with and defined by (5.3), whence where denotes the subgroup of generated by all the Then is the group of paragraph 5. Noting that is an automorphism of (since one sees that|
Remark. The theorem 7 and the preceding furnishes a completely elementary definition of the groups and but, to show the existence of the group defined in this way, one must check that the conditions (6.4) and (6.7) are satisfied, which is very long.
If an invariant subgroup of contains it coincides with
This results from the fact that, is transitive on the conjugates of on are all the conjugates of in
If is commutative, contains all the elements of the form
In effect, since is commutative, contains the derived group of so is none other than the commutator of and
If is commutative and if there exists at least one element such that the equation has a solution for all then is simple. If all is commutative, is the derived group of
Let be a nontrivial invariant subgroup of This will be doubly transitive, contains at least one transformation taking to let If is commutative, the commutator of and of this transformation is of the form If satisfies the condition of the statement, this last transformation possesse at least one fixed point outside of and it is moved to by every transformation of which preserves and which brings the fixed point to Finally, the commutator of and of (cf. (6.1)) is with However, due to the hypothesis made on when is from it is the same as The second part of the statement follows from the simplicity of and the relation
Theorem 8 has as an immediate consequence the part of Theorem 1 which concerns and We also note the following corollary.
(Suzuki) Let be a doubly transitive group of permutations of a finite set of points. If contains permutations different from the identity having two fixed points, and if, conversely, the identity is the only permutation in which has more than two fixed points, then is simple.
The subgroup of formed by the elements of the group that leave fixed a given point is a Frobenius group, so that the elements of which do not leave fixed any point other than constitutes an invariant subgroup of simply transitive on One may thus apply the preceding considerations. A transformation in which exchanges and may not have an odd cycle otherwise a power of this transformation will have more than 2 fixed points without being the identity; it follows that this transformation possesses exactly one fixed point, and thus that its square is the identity. Let and be defined as above, and two elements of and The transformations and which all exchange and are involutions. It follows that giving that is commutative, and The order of divides (since, by hypothesis, is simply transitive on the different orbits of and it is thus odd. It is a result that each element of is a square, and is thus of the for and one has in virtue of the proposition 4.
On the other hand, if there are and the equality leads to whence in other words, the map of to itself defined by is injective, thus bijective; and one is in the conditions for application of theorem 8. The corollary is thus proved.
Remark. The group over a finite field satisfies the conditions of the preceding statement.
This paragraph and the two following are directly inspired by two still unpublished articles of R. REE  and , who has kindly allowed us to make use of his manuscripts.
The automorphisms of are denoted exponentially; moreover, all the maps (and in particular the automorphisms of act on the left.
Only this case is considered by R. REE. Some of the results presented concerning the nonperfect case have only been verified hastily; we hope that they are never-the-less correct.
It is not impossible that (whence for some
The following properties are not stated by R. REE, but they are deduced very easily from his results.
One may also check by direct computation that The only point is that a bit delicate in the proof consists in establishing that, on implies but this results immediately from the relation a consequence of 4.2.1.
Remark communicated to the author by Ms FELLEGARA.
This is essentially the definition of the group of SUZUKI.
is easily calculated by equating with the two sides of (6.8) and making use of the following, easy to establish, identities:
We equate with the two sides of (6.8) and we make use of the identities:
 BARLOTTI (Adriano). – Un'estensione del teorema di Segre-Kustanheimo, Boll. Unione mat. Italiana, v. 10, 1955, p. 498-506.
 BOURBAKI (Nicolas). – Algèbre, chap. 7. – Paris, Hermanna, 1948 (Act. scient. et ind., 1044; Eléments de Mathématique, 14).
 CHEVALLEY (Claude). – Sur certains groups simples, Tôhoku math. J., Series 2, v. 7, 1955, p. 14-66.
 REE (Rimhak). – A family of simple groups associated with the simple Lie algebra of type Amer. J. of Math. (to appear).
 REE (Rimhak). – A family of simple groups associated with the simple Lie algebra of type Amer. J. of Math. (to appear).
 SEGRE (Beniamino). – Le geometrie de Galois, Annali di Mathematica, Serie 4, v. 48, 1959, p. 1-96.
 SEGRE (Beniamino). – On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arithmetica, v. 5, 1959, p. 315-332.
 STEINBERG (Robert). – Variations on a theme of Chevalley, Pacific J. of Math., v. 9, 1959, p. 875-891.
 SUZUKI (Michio). – A new type of simple groups of finite order, Proc. Nat. Acad. Sc. U.S.A., v. 46, 1960, p. 868-870.
 TITS (Jacques). – sur les analogues algèbriques des groupes semi-simples complexes, Colloque d'Algèbre supérieure [1956. Bruxelles]; p. 261-289. – Louvain, Ceuterick, 1957 (Centre belge de Recherches mathématiques).
 TITS (Jacques). – Let "formes réles" des group te type Séminaire Bourbaki, v. 10, 1957/58, no. 162, 15 p.
 TITS (Jacques). – Sur la tralité et certain groupes qui s'en déduisent. – Paris, Presses universitaires de France, 1959 (Inst. H. Et. scient. Publications mathématiques, 2; p. 14-60).
AMS Subject Classifications: Primary 05E05; Secondary 33D52.
This is a translation of the paper
Tits, Jacques, Les groupes simples de Suzuki et de Ree, Séminaire N. Bourbaki, 1960-1961, exp. no. 210, p. 65-82, http://www.numdam.org/item?id=SB\_1960-1961\_\_6\_\_65\_0
translated by Arun Ram, January 2015.