Group of type of Dregs

Groups of Chevalley

The theory was clarified by the theory of the algebraic groups and by the work of Claude Chevalley, in the middle of the Fifties, on the algebras of Dregs to the means of which the concept of group of Chevalley was isolated. Chevalley built a Base of Chevalley (a kind of integral form) for all the simple algebras of Dregs complex (or rather of their universal algebras enveloping), which can be used to define the corresponding algebraic groups on the whole . In particular, it could take their points with values in any finished body. For the algebras of Dregs $A\_n\; \backslash ,$, $B\_n\; \backslash ,$, $C\_n\; \backslash ,$, $D\_n\; \backslash ,$ this gave the well-known traditional groups, but its construction gave also the groups associated with the exceptional algebras of Dregs

$E\_6\; \backslash ,$, $E\_7\; \backslash ,$, $E\_8\; \backslash ,$, $F\_4\; \backslash ,$ and $G\_2\; \backslash ,$. (Some of those had already been built by Dickson.)

Groups of Steinberg

The construction of Chevalley did not give all the known traditional groups: it omitted the unit groups and the nonseparate orthogonal groups. Steinberg found a modification of the construction of Chevalley which gives these groups and some new families. Its construction is similar to the usual construction of the unit group starting from the general linear group. The general linear group on the complex numbers has a " automorphism of diagramme" given by taking the transposed opposite one and a " automorphism of corps" given by taking the complex conjugation. The unit group is the group of the point fixed of the product of these two automorphisms. Same manner, much of groups of Chevalley have " automorphisms of diagramme" induced by the automorphisms of their diagrams of Dynkin and the " automorphisms of corps" induced by the automorphisms of a finished body. Steinberg built families of groups by taking fixed points of a product of a diagram and an automorphism of body. Those gave the unit groups:

$\{\}^2\backslash !A\_n\; \backslash ,$ coming from the automorphism of order 2 of $A\_n\; \backslash ,$, certain additional orthogonal groups $\{\}\; ^2\; \backslash !\; D\_n\; \backslash ,$ of the automorphism of order 2 of $D\_n\; \backslash ,$ and two new series of $\{\}\; ^2\; \backslash !\; E\_6\; \backslash ,$,

Groups of Suzuki-Ree

Around 1960, Suzuki created sensation by discovering a new infinite series of groups which seemed, first of all, not connected to the known algebraic groups. Known Ree that the algebraic group $B\_2\; \backslash ,$ had an automorphism " supplémentaire" of characteristic 2 whose el square was the Automorphisme of Frobenius. It found that if a finished body of characteristic 2 has also an automorphism whose square is the application of Frobenius, then an analog of the construction of Steinberg gives the groups of Suzuki. The bodies with such an automorphism are those of order $2^\; \{2n+1\}\; \backslash ,$, and the corresponding groups are the groups of Suzuki

$\{\}^2\backslash !B\_2\; (2^\; \{2n+1\})\; =\; Suz\; (2^\; \{2n+1\})\; \backslash ,$.

(with strictly being spoken, the group Suz (2) is not counted as a group of Suzuki as it is not simple: it is the Groupe of Frobenius of order 20). Ree could find two new families similar

$\{\}^2\backslash !F\_4\; (2^\; \{2n+1\})\; \backslash ,$

and

$\{\}^2\backslash !G\_2\; (3^\; \{2n+1\})\; \backslash ,$

simple groups by using the fact that $F\_4\; \backslash ,$ and $G\_2\; \backslash ,$ have additional automorphisms of characteristics 2 and 3. (While speaking coarsely, in characteristic p , it is allowed to be unaware of the arrow on the bonds of multiplicity p in the diagram of Dynkin by taking the automorphisms of diagram). More the small group $\{\}\; ^2\; \backslash !\; F\_4\; (2^\; \{2\}\; \backslash ,$ of the type $\{\}\; ^2\; \backslash !\; F\_4\; \backslash ,$ is nonsimple, but it has a simple sub-group of index 2, called the group of Tits (named in the honor of the Belgian mathematician Jacques Tits). More the small group $\{\}\; ^3\; \backslash !\; G\_2\; (3)\; \backslash ,$ of the type $\{\}\; ^3\; \backslash !\; G\_2\; \backslash ,$ is nonsimple, but it has a sub-group normal of index 3, isomorph with $SL\_2\; (8)\; \backslash ,$. In the classification of the finished simple groups, the groups of Ree

$\{\}^2\backslash !G\_2\; (3^\; \{2n+1\})\; \backslash ,$

are those of which the structure is more diffiicle with " dégoupiller" explicitly. These groups also played a part in the discovery of the first modern sporadic group. They have centralizers of involution of the form Z /2 Z   ×   PSL ₂ ( Q ) for Q = 3 ^N , and by studying the groups with a centralizer of similar involution of form Z /2 Z   ×   PSL ₂ (5) Janko discovered the sporadic group '' J '' ₁.

Small groups of the type of Dregs

Many the more small groups in the families above have special properties not shared by the majority of the family members.

• Quelquefois more the small groups are resolvable rather than simple; for example the groups SSL ₂ (2) and SSL ₂ (3) are resolvable.

• There exists a confusing number of isomorphisms " accidentels" between various small groups of the type of Dregs (and alternate groups). For example, the groups SSL ₂ (4), PSL ₂ (5) and the group alternated on 5 points are all isomorphous.

• Some of the small groups have a Multiplicateur of Schur which larger than is envisaged. For example, the groups has _N ( Q ) have usually a multiplier of Schur of order ( N   +  1, Q   −   1), but the group has ₂ (4) has a multiplier of Schur of order 48, in the place of the value envisaged 3.

For a complete listing of these exceptions, to see the List of the finished simple groups. Many of these special properties are connected to certain simple sporadic groups. The existence of these small phenomena is not entirely a “commonplace” question; they are elsewhere considered, for example in the Théorie of homotopy.

The alternate groups behave sometimes as if they were groups of the types of Dregs on the body (non-existent) with 1 element. Some of the alternate small groups have also exceptional properties. The alternate groups have usually a Groupe of automorphism external of order 2, but the group alternated on 6 points has a group of automorphism external of order 4. The alternate groups have a multiplier of Schur of order 2, but those on 6 or 7 points have a multiplier of Schur of order 6.

Questions of notations

Unfortunately, there does not exist standard notation for the groups of the type of Dregs finished, and the literature contains dozen marking systems incompatible and confused.

• the groups of the type $A\_\; \{n-1\}\; \backslash ,$ are sometimes noted by $PSL\_n\; (Q)\; \backslash ,$ (the special linear projective group) or by $L\_n\; (Q)\; \backslash ,$.

• the groups of the type $C\_n\; \backslash ,$ are sometimes noted by $Sp\_\; \{2n\}\; (Q)\; \backslash ,$ (the group symplectic) or by $Sp\_n\; (Q)\; \backslash ,$ (in the case of authors with the particularly diabolic spirit).

• the notation for the orthogonal groups is particularly confused. Certain symbols used are $O\_n\; (Q)\; \backslash ,$, $O^\; \{-\}\; \_n\; (Q)\; \backslash ,$, $PSO\_n\; (Q)\; \backslash ,$, $\backslash \; Omega\_n\; (Q)\; \backslash ,$, but there exists so much of conventions which it is not possible to say exactly which groups correspond to it. Certain authors use a particularly vicious trap: $O\_n\; (Q)\; \backslash ,$ for a group which is not not the orthogonal group, but the simple group corresponding.

• For the groups of Steinberg, certain authors write $\{\}\; ^2\; \backslash !\; A\_n\; (q^2)\; \backslash ,$ (and so on) for the group that other authors indicate by $\{\}\; ^2\; \backslash !\; A\_n\; (Q)\; \backslash ,$. The problem is that there exist two implied bodies, one of order $q^2\; \backslash ,$ and its fixed body of order Q and the readers has different ideas on that which should be included in the notation. Convention " $\{\}^2\backslash !A\_n\; (q^2)\; \backslash ,$" is more logical and in conformity, but convention " $\{\}^2\backslash !A\_n\; (Q)\; \backslash ,$" is most common by far.

• the authors so differ from the groups such as $A\_n\; (Q)\; \backslash ,$ is the groups of points to values in the simple group or the simply connected algebraic group. For example, $A\_n\; (Q)\; \backslash ,$ can mean is the linear special group $SL\_\; \{n+1\}\; (Q)\; \backslash ,$ or the linear special projective group $PSL\_\; \{n+1\}\; (Q)\; \backslash ,$. Thus $\{\}\; ^2\; \backslash !\; A\_n\; (2^2)\; \backslash ,$ can be one of the 4 different groups, depend on the author.

Advanced readings

A standard reference:

• Simple Groups off Standard Dregs by Roger W. Casing, ISBN 0-471-50683-4

The traditional groups are described in

• geometry of the traditional groups by Jean Dieudonné