The simple groups of Suzuki and Ree

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 14 January 2015

The groups $B_2^{*},$ $G_2^{*}$ and $F_4^{*}$ $\left({}^1\right)\text{.}$

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 $K$ be a field of characteristic $p$ $\text{(}=$ or $\ne 0\text{),}$ $𝔤$ a complex semisimple Lie algebra, $\mathrm{\Sigma }$ a system of simple roots of $𝔤,$ $\mathrm{\Delta }$ the Dynkin diagram of $𝔤$ (in the following we will assume that we can distinguish the roots of different lengths), $G$ the "Tôhoku group" [3] of type $𝔤$ over $K\text{,}$ and let $x_a\left(t\right)\text{,}$ ${𝒳}_a=\left\{x_a\left(t\right)\right\}$ and $h\left(\chi \right)$ defined as in [3]. To every automorphism $\pi$ of $\mathrm{\Delta }$ there corresponds an isometric permutation of $\mathrm{\Sigma }$ which extends to an isometric permutation of the set of all roots, this will also be denoted by $\pi \text{.}$ We will denote ${\alpha }_{\pi }$ the automorphism of $G$ defined by $$x_a\left(t\right)⟼x_{\pi \left(a\right)}\left(t\right)(a\in \mathrm{\Sigma })$$ $$h\left(\chi \right)⟼h\left(\chi \prime \right)with\chi \prime \left(\pi \left(a\right)\right)=\chi \left(a\right)\text{.}$$ For every root $a\text{,}$ this automorphism transforms ${𝒳}_a$ into ${𝒳}_{\pi \left(a\right)}\text{.}$ By abuse of language, the automorphisms of $G$ which are the product of ${\alpha }_{\pi }$ by an inner automorphism will be called birational.

To each endomorphism $\theta$ of $K\text{,}$ on may associate an endomorphism ${\alpha }_{\theta }$ of $G$ defined by $$x_a\left(t\right)⟼x_a\left(t^{\theta }\right),h_{\chi }⟼h\left({\chi }^{\theta }\right)\left({}^2\right)\text{.}$$ Every endomorphism which is a product of a birational endomrphism with ${\alpha }_{\theta }$ will be said to belong to $\theta \text{.}$

If $K$ is a galois extension of the group $\mathrm{\Gamma }$ of a field $L\text{,}$ an $L\text{-form}$ of $G$ may be defined as the group of elements fixed by an automorphism group $\lambda \left(\mathrm{\Gamma }\right)$ of $G\text{,}$ the image of $\mathrm{\Gamma }$ by a monomorphism $\lambda :\mathrm{\Gamma }\to \text{Aut}\left(G\right)$ (the group of automorphisms of $G\text{)}$ such that $\lambda \left(\gamma \right)$ belongs??? to $\gamma$ for every $\gamma \in \mathrm{\Gamma }\text{.}$ In particular, to each monomorphism $\mu :\mathrm{\Gamma }\to \text{Aut}\left(\mathrm{\Delta }\right),$ one may make a corresponding $L\text{-form}$ of $G\text{,}$ with the help of the monomorphism $\lambda$ defined by $$\lambda \left(\gamma \right)={\alpha }_{u\left(\gamma \right)}·{\alpha }_{\gamma }\text{.}$$ The formes of $D_4$ and $E_6\text{,}$ which have been considered by D. HERTZIG, R. STEINBERG [8] and the author [11] and [12], are of this type.

If one ignores the length of roots, three Dynkin diagrams acquire an automorphism $\pi$ of order 2; these are the diagrams of $B_2\text{,}$ $G_2$ and $F_4\text{.}$ To this automorphism, there corresponds a permutation of $\mathrm{\Sigma }$ 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 $\pi \text{.}$ In certain special cases, one may also associate an automorphism, or more generally an endomorphism of $G\text{,}$ no longer "birational" but only "rational". More precisely, if $𝔤=B_2$ or $F_4$ (resp. $G_2\text{)}$ and $p=2$ (resp. 3), a simple calculation shows that there exists an endomorphism ${\alpha }_{\pi }$ of $G$ defined by $$\begin{array}{l}{\displaystyle x_a\left(t\right)⟼x_{\pi \left(a\right)}\left(t^{\ell \left(a\right)}\right)(a\in \mathrm{\Sigma })}\\ {\displaystyle h\left(\chi \right)⟼h\left(\chi \prime \right)\text{with}\chi \prime \left(\pi \left(a\right)\right)=\chi {\left(a\right)}^{\ell \left(a\right)},}\end{array}$$ where $\ell \left(a\right)=1$ or $p$ according to whether $a$ is a "long" or "short" root. It is clear that ${\alpha }_{\pi }^2={\alpha }_{\phi }$ where $\phi :k\mapsto k^p$ is the Frobenius of $K\text{.}$ Assume that $K$ possesses an endomorphism $\sigma$ such that ${\sigma }^2=\phi \text{.}$ Then $\alpha ={\alpha }_{\pi }^{-1}·{\alpha }_{\sigma }$ is an involutive automorphism $\text{(}{\alpha }^2=1\text{)}$ of the group ${\alpha }_{\sigma }^{-1}({\alpha }_{\sigma }\left(G\right)\cap {\alpha }_{\pi }\left(G\right))\text{.}$ In particular, if $K$ is perfect, $\phi ,$ $\sigma \text{,}$ ${\sigma }_{\pi }$ and ${\alpha }_{\sigma }$ are surjective, and $\alpha$ is an automorphism of $G$ $\left({}^3\right)\text{.}$ In all cases, we denote $G^{*}$ $\text{(}=B_2^{*},$ $G_2^{*}$ or $F_4^{*}\text{)}$ the group of elements $g$ of $G$ such that ${\alpha }_{\pi }\left(g\right)={\alpha }_{\sigma }\left(g\right)$ (elements invariant by $\alpha \text{).}$

Let $U_{+}$ (resp. $U_{-}\text{)}$ be the subgroup of $G$ generated by the ${𝒳}_a$ (resp. the ${𝒳}_{-a}\text{)}$ $\text{(}a\in \mathrm{\Sigma }\text{),}$ and $U_{+}^{*}=U_{+}\cap G^{*}$ (resp. $U_{-}^{*}=U_{-}\cap G^{*}\text{).}$ Then one has the following theorem:

If $K$ has more than $p$ $\text{(}=2$ or $3\text{)}$ elements, the group generated by $U_{+}^{*}$ and $U_{-}^{*},$ which is non other than the derived group $G^{*}\prime$ of $G^{*},$ is simple. Furthermore, $F_4^{*}=F_4^{*}\prime ,$ $B_2^{*}/B_2^{*}\prime \cong K^{*}/{K^{*}}^{\sigma }$ (in particular, $B_2^{*}=B_2^{*}\prime$ if $K$ is perfect, and $G_2^{*}/G_2^{*}\prime \cong K^{*}/M,$ where $M$ is a subgroup of $K^{*}$ which contains ${K^{*}}^2$ and $-1$ $\left({}^4\right)\text{.}$

The part of this theorem which concerns $B_2^{*}$ and $G_2^{*}$ will be proved later, after another slightly different definition of these groups. For the case of $F_4^{*},$ we refer to [5].

Cellular decompositions

The notations are as in paragraph 1, with $W$ the Weyl group of $𝔤$ $\text{(}=B_2,$ $F_4\text{,}$ or $G_2\text{),}$ $\omega \left(w\right)$ ($w\in W\text{)}$ defined as in [3], $H$ the group formed by the $h\left(\chi \right)\text{,}$ $\mathrm{\Omega }$ the group generated by $H$ and the $\omega \left(w\right),$ $\varpi :\mathrm{\Omega }\to W$ the homomorphism with kernel $H$ defined by $\omega \left(w\right)\mapsto w,$ $W^{*}$ the group of elements of $W$ invariant by $\pi \text{,}$ $H^{*}=H\cap G^{*},$ $H_0^{*}=H\cap G^{*}\prime ,$ ${\mathrm{\Omega }}^{*}=\mathrm{\Omega }\cap G^{*},$ ${\mathrm{\Omega }}_0^{*}=\mathrm{\Omega }\cap G^{*}\prime \text{.}$

It is clear that $\varpi \left({\mathrm{\Omega }}^{*}\right)\subseteq W^{*}\text{.}$ In fact, we have the following theorem:

$\varpi \left({\mathrm{\Omega }}^{*}\right)=\varpi \left({\mathrm{\Omega }}_0^{*}\right)=W^{*}\text{.}$ Alternatively, ${\mathrm{\Omega }}^{*}/H^{*}\cong {\mathrm{\Omega }}_0^{*}/H_0^{*}\cong W^{*}\text{.}$

The proof of this is done by explicitly constructing for each $w\in W^{*}$ an $\omega \left(w\right)$ in $G^{*}\prime$ (the three cases, $B_2\text{,}$ $F_4\text{,}$ $G_2$ are treated separately). In the following we always assume that the $\omega \left(w\right)$ $\text{(}w\in W^{*}\text{)}$ are chosen in this fashion.

Given $w\in W\text{,}$ we put $U_{+}^w=U_{+}\cap \omega \left(w\right)·U_{-}·\omega {\left(w\right)}^{-1},$ and, when $w\in W^{*},$ $U_{+}^{*}=U_{+}\cap \omega \left(w\right)·U_{-}^{*}·\omega {\left(w\right)}^{-1}\text{.}$ Every element $g\in G$ may be put, in one and only one fashion, in the form of a product $u\prime ·\omega ·u,$ with $u\in U_{+},$ $\omega \in \mathrm{\Omega }$ and $u\prime \in U_{+}^{\varpi \left(w\right)}$ (theorem 2 of [3]). On the other hand, ${\alpha }_{\pi }$ and ${\alpha }_{\sigma }$ sends each of the groups $H\text{,}$ $U_{+},$ $U_{-}$ and $\mathrm{\Omega }$ into itself (for $H\text{,}$ $U_{+}$ and $U_{-}$ this is evident; for $\mathrm{\Omega },$ this is because $\mathrm{\Omega }$ is the normalizer of $H\text{,}$ and $H$ is the centralizer of ${\alpha }_{\pi }\left(H\right)$ and of ${\alpha }_{\sigma }\left(H\right)\text{)}$ it follows that if $g=u\prime ·\omega ·u$ belongs to $G^{*},$ then the same is true of $u\text{,}$ $u\prime$ and $\omega \text{.}$ Thus,

Every element of $G^{*}$ (resp. $G^{*}\prime \text{)}$ may be put, in one and only one fashion, in the form of a product $u\prime ·\omega ·u,$ with $u\in U_{+}^{*},$ $\omega \in {\mathrm{\Omega }}^{*}$ (resp. ${\mathrm{\Omega }}_0^{*}\text{)}$ and $u\prime \in U_{+}^{*\varpi \left(\omega \right)}\text{.}$

We will also give, for each of the groups $B_2^{*},$ $G_2^{*}$ and $F_4^{*},$ some indications of the structure of $W^{*},$ $U_{+}^{*},$ $H^{*}$ and $H_0^{*}\text{.}$

Case $𝔤=B_2$ (resp. $G_2\text{).}$ $W^{*}$ is composed then of the identity and the operation $w_0$ which transforms each root to its opposite. $\omega \left(w_0\right)$ exchanges $U_{+}^{*}$ and $U_{-}^{*},$ then $G^{*}=U_{+}^{*}·H^{*}·\omega \left(w_0\right)·U_{+}^{*}\cap U_{+}^{*}·H^{*},$ by virtue of theorem 3. It follows that $G^{*}$ is doubly transitive on the space $\mathrm{\Gamma }=G^{*}/U_{+}^{*}·H^{*},$ of left cosets of $U_{+}^{*}·H^{*}$ in $G^{*}\text{.}$ In effect, one may canonically identify $\mathrm{\Gamma }$ with the union $U_{+}^{*}\cup \left\{\mathbf{\infty }\right\}$ of $U_{+}^{*}$ and a point $\mathbf{\infty },$ the image of the identity element, and the subgroup $U_{+}^{*}$ of $G^{*}$ acts transitively on itself and preserves $\mathbf{\infty }\text{.}$ We will rediscover later (paragraphs 4, 5, 6) this doubly transitive representation of $G^{*}\text{.}$ The subgroup $H^{*}$ is isomorphic to the multiplicative group $K^{*}$ of $K$ (resp. to a subgroup $M$ satisfying the conditions of theorem 1). $U_{+}^{*}$ is nilpotent of class 1 (resp. 3) and the quotients of its ascending central series are all isomorphic to the additive group $K^{+}$ (the precise structure of $U_{+}^{*}$ will be given in paragraph 6). If one denotes the simple roots by $a$ and $b\text{,}$ with $a$ the short root, one has, with an appropriate normalization of the $x_r\left(t\right)\text{,}$ $$\begin{array}{rcl}{\displaystyle U_{+}^{*}}& =& {\displaystyle \{x_a\left(t\right)·x_b\left(t^{\sigma }\right)·x_{a+b}\left(u\right)·x_{2a+b}(t^{\sigma +2}+u^{\sigma })\}\text{(}t,u\in K\text{)}}\\ {\displaystyle \text{(resp.}}& & {\displaystyle \phantom{\{}{x}_a\left(t\right)·x_b\left(t^{\sigma }\right)·x_{a+b}\left(u\right)·x_{2a+b}\left(v\right)·x_{3a+b}(u^{\sigma }-t^{\sigma +2})·x_{3a+2b}(v^{\sigma }-t^{2\sigma +2})\text{(}t,u,v\in K\text{)).}}\end{array}$$

Case $𝔤=F_4$ $W^{*}$ is then a dihedral group of order 16. $H^{*}$ is isomorphic to $K^{*}\times K^{*}\text{.}$ Further $\left({}^5\right),$ there exist subgroups ${𝒳}_i$ and ${𝒴}_i$ $\text{(}i=\pm 1,\pm 2,\pm 3,\pm 4\text{),}$ respectively isomorphic to $K^{+}$ and to the "unipotent" $U_{+}^{*}$ of $B_2^{*}\text{,}$ and having properties completely analogous to those ot the ${𝒳}_r$ of the Tôhoku groups, so that, for example:

	${𝒳}_i$ and ${𝒴}_i$ are invariant under $R^{*}\text{;}$
	the inner automorphisms corresponding to elements of $\mathrm{\Omega }$ permute the ${𝒳}_i$ and the ${𝒴}_i$ 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 $F_4^{*}$ is a linear sextuple, which somewhat breaks the monotonie $J\text{??????);}$
	If $j$ groups ${𝒵}_1,\dots ,{𝒵}_j$ are chosen from the ${𝒳}_i$ and ${𝒴}_i\text{,}$ corresponding to the half lines of figure 1 contained in an acute algebra ("convex cone") $\mathrm{\Theta }\text{,}$ every element of the group $U_{\mathrm{\Theta }}$ which they generate can be put, in a unique fashion, in the form of a product $z_1·z_2\cdots z_j,$ with $z_m\in {𝒵}_m\text{;}$
	if $\mathrm{\Theta }$ and $\mathrm{\Theta }\prime$ are two acute angles, $U_{\mathrm{\Theta }\cap \mathrm{\Theta }\prime }=U_{\mathrm{\Theta }}\cap U_{\mathrm{\Theta }\prime }\text{;}$
	$U_{+}$ is generated by the ${𝒳}_i\text{,}$ ${𝒴}_i$ $\text{(}i>0\text{);}$
	for every $i\text{,}$ ${𝒳}_i$ and ${𝒳}_{-i}$ (resp. ${𝒴}_i$ and ${𝒴}_{-i}\text{)}$ generate a group isomorphic to ${PSL}_2\left(K\right)$ (resp. to $B_2^{*}\text{).}$
	if $G_x^{*}$ (resp. $G_y^{*}\text{)}$ denotes the group generated by $U_{+}$ and ${𝒳}_{-i}$ (resp. ${𝒴}_{-i}\text{),}$ the sets $G^{*}/G_x^{*}$ and $G^{*}/G_y^{*}$ constitute, with the usual incidence relation [10], a generalised octagon (cf. [12], Appendix);
	one has also a "geometric interpretation" of $F_4^{*}$ as the group of automorphisms of this octagon;

etc.

Finite groups

Let $K$ be the finite field with $q=2^n$ (resp. $3^n\text{)}$ elements. We see easily that the Frobenius $x\mapsto x^2$ (resp. $x^3\text{)}$ possesses a square root, d'aillers?????? unique, if and only if $n=2m+1\text{.}$ In this case, one may define the finite groups $B_2^{*}$ and $F_4^{*}$ (resp. $G_2^{*}\text{),}$ which are simple when $m>0,$ by virtue of Theorem 3 (in the case $G_2^{*}\text{,}$ it is remarked that, because of the hypothesis made on $n\text{,}$ $-1$ is not a square in $K\text{,}$ from which it results that $-1$ and ${\left(K^{*}\right)}^2$ generate $K^{*}\text{).}$ The orders $o\left(G^{*}\right)$ are easily calculated from the results of paragraph 2, and one finds $$\begin{array}{rcl}{\displaystyle o\left(B_2^{*}\right)}& =& {\displaystyle q^2·(q^2+1)·(q-1),}\\ {\displaystyle o\left(G_2^{*}\right)}& =& {\displaystyle q^3·(q^3+1)·(q-1),}\\ {\displaystyle o\left(F_4^{*}\right)}& =& {\displaystyle q^{12}·(q^6+1)·(q^4-1)·(q^3+1)·(q-1)\text{.}}\end{array}$$

One notes that $o\left(B_2^{*}\right)$ is not divisible by $3,$ a property that is not shared by any other known simple group. R. REE notes that one may show that the $G_2^{*}$ are also new simple groups by using the fact that $o\left(G_2^{*}\right)$ is divisible by $8$ and not by $16\text{.}$ Finally one may follow the method of Artin to give an analogous result for $F_4^{*}\text{.}$ The $B_2^{*}$ are the groups of SUZUKI [9] (cf. number 4.3).

A geometric definition of the groups $B_2^{*}\text{.}$

Duality in the "linear complex" in characteristic 2.

We will employ in many instances the following abbreviated notation: "A projective space $\mathrm{\Xi }({\xi }_0,{\xi }_1,\dots )\text{"}=$ "A projective space $\mathrm{\Xi }$ in which is given a homogeneous system of coordinates ${\xi }_0,{\xi }_1,\dots \text{".}$ On the other hand, a variety will always be assimilated from the set of its rational points over $K\text{.}$

Let $K$ be a field of characteristic $p$ and $E(x_0,x_1,x_2,x_3)$ a 3-dimensional projective space over $K\text{.}$ The lines of $E$ may be specified in terms of their "Pücker coordinates" $p_{ij}$ which satisfy the relation $p_{01}·p_{23}+p_{02}·p_{31}+p_{03}·p_{12}=0\text{.}$ We consider the set ("linear complex") $L$ of lines such that $p_{01}=p_{23}\text{.}$ These are represented, in a 4-dimensional projective space $D(p_{01},p_{02},p_{03},p_{12},p_{31}),$ by the points of a hyperquadric $V\text{,}$ with equation $$p_{01}^2+p_{02}·p_{31}+p_{03}·p_{12}=0\text{.}$$ Now we assume that the characteristic $p$ of $K$ is equal to $2\text{.}$

The hyperplanes tangent to $V$ then meet at a point, with coordinates $p_{02}=p_{03}=p_{12}=p_{31}=0,$ and this permits the definition (by "projection" from this point), a "rational" injection of $V$ into a projective space $F(y_0,y_1,y_2,y_3),$ by $$(p_{01},p_{02},p_{03},p_{12},p_{31})⟼(y_0=p_{02},y_1=p_{31},y_2=p_{03},y_3=p_{12})\text{.}$$

The Plücker coordinates of the lines of $F$ will be denoted $q_{ij}\text{.}$ To the set of lines of $L$ which contain a given point $(x_0,\dots ,x_3)$ of $E$ correspond a line of $V\text{,}$ thus a line of $F$ for which the Plücker coordinates are, after an elementary calculation, $$q_{01}=q_{23}=x_0·x_1+x_2·x_3,q_{02}=x_0^2,q_{03}=x_2^2,q_{31}=x_1^2,q_{12}=x_3^2,$$ and which, in particular, belong to the set of lines of $M$ with equation $q_{01}=q_{23}\text{.}$ In this way a sort of duality is established between $(E,L)$ and $(F,M),$ formed by two injections $\delta :L\to F$ and $\delta \prime :E\to M,$ such that if $\text{x}$ and $d$ denote, respectively, a point of $E$ and a line belonging to $L\text{,}$ the properties of $\text{x}\in d$ and $\delta \left(d\right)\in \delta \prime \left(\text{x}\right)$ are equivalent.

We will denote by $G(E,L)$ (resp. $G(F,M)$ the group, isomorphic to $P\mathrm{\Gamma }{Sp}_4\left(X\right)=B_2$ of Tôhoku, of projectivities of $E$ preserving $L$ (resp. of projectivities of $F$ preserving $M\text{).}$ The "duality" between $(E,L)$ and $(F,M)$ "induces" a monomorphism $\delta :G(E,L)\to G(F,M),$ defined by ${\delta }^{*}\left(g\right)\left(\delta \prime \left(d\right)\right)=\delta \prime \left(g\left(d\right)\right),$ for each $g\in G(E,L)$ and each line $d\in L\text{.}$

The "polarity" $f$

Now we assume that the Frobenius of $K$ possesses a square root $\sigma \text{,}$ and we consider the semilinear map $f:E\to F$ defined by $$(x_0,\dots ,x_3)⟼(y_0=x_0^{\sigma },\dots ,y_3=x_3^{\sigma })\text{.}$$ From number 4.1, it follows immediately that $\text{x}$ and $\text{x}\prime$ are points of $E\text{,}$ the relation $\text{"}f\left(x\right)$ is in the line $\delta \prime \left(x\prime \right)\text{"}$ is symmetric; thus $f$ functions as a sort of polarity. We will denote by $\mathrm{\Gamma }$ the set of points $\text{<mi>x</mi>}$ such that the point $f\left(x\right)$ is in its "polar" $\delta \prime \left(\text{x}\right)\text{.}$

We introduce now, in $E$ without the plane $x_0=0$ ("plane at infinity"), with the nonhomogeneous coordinates $x=x_2/x_0,$ $y=x_3/x_0,$ $z=x_1/x_0\text{.}$ An elementary calculation shows that $\mathrm{\Gamma }$ is composed of the point $x_0=x_2=x_3=0,$ which we denote by $\mathbf{\infty },$ and the set $\mathrm{\Gamma }-\left\{\mathbf{\infty }\right\},$ completely "at finite distance", of the equation $$\begin{array}{cc}{\displaystyle z=x·y+x^{\sigma +2}+y^{\sigma }\text{.}}& \text{(4.2.1)}\end{array}$$

The groups $G^{*}$ and $G_0^{*}\text{.}$

We see easily that there exists a monomorphism $f^{*}:G(E,L)\to G(F,M)$ defined by $f^{*}\left(g\right)\left(f\left(\text{x}\right)\right)=f\left(g\text{x}\right),$ for all $\text{x}\in E$ and $g\in G\text{.}$

We will denote by $G^{*}\left(f\right)$ the group of elements $g\in G(E,L)$ such that $f^{*}\left(g\right)={\delta }^{*}\left(g\right)$ (group of projectivities "convserving $f\text{");}$ $G^{*}\left(f\right)$ is not other than the group $B_2^{*}$ (the link between the two definitions is apparent). It is clear that $G^{*}\left(f\right)$ is contained in the group of all projectivities of $E$ which preserve the set $\mathrm{\Gamma }\text{,}$ this will be denote by $G^{*}$ (in fact, $G^{*}\left(f\right)=G^{*}\text{:}$ we will see later). The group of elements of $G^{*}\text{,}$ where the determinant is a 4th power, will be denoted $G_0^{*}\text{.}$

The projectivities $$t_{a,b}:(x,y,z)⟼(x+a,y+b+a^{\sigma }x,z+ab+a^{\sigma +2}+b^{\sigma }+ay+a^{\sigma +1}x+bx)\text{(}a,b\in K\text{),}$$ $$h_k:(x,y,z)⟼(kx,k^{\sigma +1}y,k^{\sigma +2}z)\text{(}k\in K^{*}\text{),}$$ $$\omega (x_0,x_1,x_2,x_3)⟼(x_1,x_0,x_3,x_2),$$ are in $G^{*}\text{.}$ For $t_{a,b}$ and $h_k\text{,}$ this follows from an easy calculation; for the projectivity $\omega \text{,}$ it is immediate that it is in $G^{*}\left(f\right)$ $\left({}^6\right)\text{.}$ Further, $t_{a,b}$ and $\omega \in G_0^{*},$ giving that $h_k\in G_0^{*}$ if and only if $k\in {K^{*}}^{\sigma }\text{.}$ Finally, one verifies easily that $\left\{t_{a,b}\right\}=T^{*}$ and $\left\{h_k\right\}=H^{*}$ are groups.

If $K$ has more than two elements, the $h_k$ are the only elements of $G^{*}$ that preserve the points $\mathbf{\infty }$ and $x=y=z=0\text{.}$

		Proof.
	A projectitivy that preserves $\mathbf{\infty }$ and $\mathrm{\Gamma }$ preserves the plane $x_0=0,$ which is in effect the only plane containing $\mathbf{\infty }$ where the intersection with $\mathrm{\Gamma }$ is reduced to this point. It follows that a projectivity that preserves $\mathrm{\Gamma }\text{,}$ $\mathbf{\infty }$ and $x=y=z=0$ is a linear transformation represented by a linear and homogeneous substitution on $x,y,z\text{.}$ Taking into account the fact that $\mathrm{\Gamma }$ and $\mathbf{\infty }$ are preserved, we see that if we make the substitution in the function $x+xy+x^{\sigma +2}+y^{\sigma },$ this must be multiplication by a constant, but it is also the same as the "homogeneous components" $z,xy,x^{\sigma +2},y^{\sigma },$ because of the linear independence of multiplicative endomorphisms of $K$ (cf. [2], paragraph 7, no. 5) (we use here the fact that $K^{*}\ne F_2,$ because one must assume that the endomorphisms $1,$ $2,$ $\sigma$ and $\sigma +2$ are all different). It follows that the projectivity in question is of the form $(x,y,z)\mapsto (k_{1}x,k_{2}y,k_{3}z)$ and the proof is completed by a simple calculation. $\square$

The group $G_0^{*}$ (and a fortiori $G^{*}\text{)}$ is doubly transitive on $\mathrm{\Gamma }\text{.}$ If $K$ has more than 2 elements, $G^{*}$ (resp. $G_0^{*}\text{)}$ is generated by the projectivities $t_{a,b}$ $\text{(}a,b\in K\text{),}$ $h_k$ $\text{(}k\in K^{*}\text{)}$ (resp. ${K^{*}}^{\sigma }\text{)}$ and $\omega$ $\left({}^8\right)\text{.}$ Every element of $G^{*}$ may be put, in one and only one fashion, in one of the two forms $t_{a,b}·h_k$ or $t_{a,b}·h_k·\omega ·t_{c,d}$ $\text{(}a,b,c,d\in K,$ $k\in K^{*}\text{)}$ Finally, $G^{*}/G_0^{*}\cong K^{*}/{K^{*}}^{\sigma }\text{.}$

		Proof.
	The group generated by the $t_{a,b},$ the $h_k$ $\text{(}k\in {K^{*}}^{\sigma }\text{)}$ and $\omega$ is doubly transitive on $\mathrm{\Gamma }\text{;}$ in effect, the subgroup $T^{*}\text{,}$ which preserves $\mathbf{\infty },$ is transitive on $\mathrm{\Gamma }-\left\{\mathbf{\infty }\right\},$ and $\omega$ "moves" $\mathbf{\infty }\text{.}$ The theorem is thus an immediate consequence of lemma 1 and the fact that $T^{*}$ is simply transitive on $\mathrm{\Gamma }-\left\{\mathbf{\infty }\right\}\text{.}$ $\square$

$G^{*}\left(f\right)=G^{*},$

It suffices to verify that $t_{a,b},$ $h_k$ and $\omega$ belong to $G^{*}\left(f\right)\text{.}$

Remark. We will note that theorem 4 has been established solely on the basis of the analytic definition (4.2.1) of $\mathrm{\Gamma }\text{,}$ without making use of $f$ (if one takes into account the note $\left({}^7\right)\text{.}$

Properties of $\mathrm{\Gamma }\text{.}$ Calottes.

The equation (4.2.1) shows that every line passing through $\mathbf{\infty }$ and not contained in the plane $x_0=0$ meets $\mathrm{\Gamma }$ in one and only one point. Taking account of the "homogeneity" of $\mathrm{\Gamma }$ (Theorem 4), one deduces immediately that

Every line in $E$ meets $\mathrm{\Gamma }$ in 0, 1 or 2 points. The lines passing through a given point determine the set $\mathrm{\Gamma }$ and those that do not meet any other (tangents) form a plane.

When $K={𝔽}_q$ is the finite field with $q=2^{2m+1}$ elements, $\mathrm{\Gamma }$ is formed by $q^2+1$ points.

We denote by $E_{r,q}$ the projective space of dimension $r$ over ${𝔽}_q$ $\text{(}q=p^n,$ for some $p\text{).}$ A set of $N$ points of this space is called an $N\text{-calotte}$ 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 [1], [6], [7]). They have given, in particular, that

(Segre-Barlotti) If $p\ne 2,$ a $(q+1)\text{-calotte}$ in $E_{2,q}$ is a conic, a $(q^2+1)\text{-calotte}$ in $E_{3,q}$ is a quadric, and there does not exist a $(q^{r-1}+1)\text{-calotte}$ in $E_{r,q}$ for $r>3\text{.}$

In characteristic 2, the situation is more complicated. In $E_{2,q}$ there exist $(q+1)\text{-calottes}$ which are not conics (for example the set formed by the $q$ points of a conic and the meeting point of the tangents to it). The example of $\mathrm{\Gamma }$ shows that there exist also, in $E_{3,q},$ $(q^2+1)\text{-calottes}$ which are not quadrics (the example given by B. Segre in [7] for $q=8$ is a special case of this one $\left({}^7\right)\text{.}$ We mention also the following result (unpublished) of the author:

Let $\mathrm{\Gamma }$ be a $(q^{r-1}+1)\text{-calotte}$ in $E_{r,q}$ $\text{(}q=2^n\text{).}$ Assume that there exists on $\mathrm{\Gamma }$ a point $\mathbf{\infty }$ such that the group of projectivities preserving $\mathrm{\Gamma }$ and this point possesses an invariant subgroup which is simply transitive on $\mathrm{\Gamma }-\mathbf{\infty },$ and at least one cyclic subgroup of order $q-1\text{.}$ Then $r=2$ or $3\text{.}$ If $r=2,$ $\mathrm{\Gamma }$ is, in a system of affine coordinates $x,y$ appropriately chosen, a "curve" with equation $y=x^{\sigma }$ completed by the "point at infinity on one of the coordinate axes", $\sigma$ denotes any automorphism of $K$ having $1$ as its unique fixed point. If $r=3,$ $\mathrm{\Gamma }$ is a quadric, it is the set defined in section 4.2.

A geometric definition of $G_2^{*}$

For $G_2$ 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 $K$ be a field of characteristic 3, $p(x_{*},x_0,x_1,x_2,x_{0\prime },x_{1\prime },x_{2\prime })$ a 6 dimensional projective space over $K\text{,}$ $E$ the hyperquadric with equation $x_{*}^2+\sum _{i=0}^2{x}_i·x_{i\prime }=0,$ $L$ the set of lines (all belonging to $E\text{)}$ defined by the equations $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle {p_{*}}_i+p_{(i+1)\prime (i+2)\prime }}& =& {\displaystyle 0,}\\ {\displaystyle {p_{*}}_{i\prime }+p_{(i+1)(i+2)}}& =& {\displaystyle 0,}\\ {\displaystyle \sum _{i=0}^2{p}_{ii\prime }}& =& {\displaystyle 0,}\end{array}\text{(}i=0,1,2\text{; the indices taken mod}\hspace{0.17em}3\text{),}}& \text{(5.1)}\end{array}$$ $G(E,L)$ the group of projectivities of $P$ reserving $L$ (thus also $E\text{)}$ (Tôhoku group $G_2$ over $K\text{)}$ and $V$ the 5-dimensional variety representing $L$ in the 13-dimensional projective space $D({p_{*}}_i,{p_{*}}_{i\prime },p_{ij\prime }(\sum p_{ii\prime }=0))\text{.}$

On $V\text{,}$ ${p_{*}^3}_i,$ ${p_{*}^3}_{i\prime },$ and $p_{ii\prime }^3$ are polynomials in the $p_{ij\prime }$ $\text{(}i\ne j\text{)}$ and $p_{00\prime }-p_{11\prime }$ $\text{(}=p_{11\prime }-p_{22\prime }=\cdots \text{),}$ which also satisfy the relation $${(p_{00\prime }-p_{11\prime })}^2+p_{01\prime }·p_{10\prime }+p_{12\prime }·p_{21\prime }+p_{20\prime }·p_{02\prime }=0\text{.}$$ This permits the definition of a "rational" injection ("projection" from the linear variety $\mathrm{\Lambda }\equiv p_{ij\prime }=p_{00\prime }-p_{11\prime }=0$ $\text{(}i\ne j\prime \text{)}$ $$\begin{array}{cc}{\displaystyle ({p_{*}}_i,{p_{*}}_{i\prime },p_{ij\prime })⟼(y_{*}=p_{00\prime }-p_{11\prime },y_i=p_{(i+1)i\prime },y_{i\prime }=p_{i(i+1)\prime })}& \text{(5.2)}\end{array}$$ of $V$ in a hyperquadric $F\equiv y_{*}^2+\sum y_i·y_{i\prime }=0$ of a 6-dimensional projective space $Q(y_{*},y_i,y_{i\prime })\text{.}$ We will denote by ${q_{*}}_i,$ ${q_{*}}_{i\prime },$ etc., the Plücker coordinates of the lines of $Q\text{.}$ If $(x_{*},x_i,x_{i\prime })$ is a point of $E\text{,}$ the set of lines of $L$ which pass through the point has image in $Q$ (by the transfer of (5.2)) which is a line belonging to the set $M$ defined by the relations (5.1) where one replaces $p$ by $q\text{,}$ and such that $$q_{00\prime }-q_{11\prime }=x_{*}^3,q_{(i+1)i\prime }=x_i^3,q_{i(i+1)\prime }=x_{i\prime }^3\text{.}$$ In this way we find two injections $\delta :L\to F$ and $\delta \prime :E\to M$ establishing a sort of duality between $(E,L)$ and $(F,M)\text{.}$ Then one may, exactly as in paragraph 4, define the group $G(F,M),$ the monomorphism ${\delta }^{*}:G(E,L)\to G(F,M)$ (where the existence results immediately from the fact, easily vareifiable, that the projectivities of $D$ induced by the elements of $G(E,L)$ leave invariant $\mathrm{\Lambda }\text{),}$ and, if the Frobenius $K$ possesses a square root $\sigma \text{,}$ the "polarity" $$f:(x_{*},x_i,x_{i\prime })⟼(y_{*}=x_{*}^{\sigma },y_i=x_{*}^{\sigma },y_{i\prime }=x_{i\prime }^{\sigma }),$$ the monomorphism $f^{*}:G(E,L)\to G(F,M),$ the set $\mathrm{\Gamma }\text{,}$ the groups $G^{*}\left(f\right)$ and $G^{*}\text{.}$ The point $\mathbf{\infty }$ with coordinates $x_{*}=x_0=x_1=x_{i\prime }=0$ belongs to $\mathrm{\Gamma }$ and the set $\mathrm{\Gamma }-\left\{\mathbf{\infty }\right\},$ which is completely disjoint from the hyperplane $x_{2\prime }=0,$ has for equations, in non-homogeneous coordinates $(x,y,z,u,v,w)=(x_1/x_{2\prime },x_0/x_{2\prime },x_{*}/x_{2\prime },x_{0\prime }/x_{2\prime },x_{1\prime }/x_{2\prime },x_2/x_{2\prime }),$ $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle u}& =& {\displaystyle x^{2}y-xz+y^{\sigma }-x^{\sigma +3},}\\ {\displaystyle v}& =& {\displaystyle x^{\sigma }{y}^{\sigma }-z^{\sigma }+x{y}^2+yu-x^{2\sigma +3},}\\ {\displaystyle w}& =& {\displaystyle x{z}^{\sigma }-x^{\sigma +1}y+x^{\sigma +3}y+x^2{y}^2-y^{\sigma +1}-z^2+x^{2\sigma +4}\text{.}}\end{array}}& \text{(5.3)}\end{array}$$ The following projectivities belong to $G\text{:}$ $$t_{a,b,c}:(x,y,z,\dots )⟼(x+a,y+a^{\sigma }+b,z-ay+bx-a^{\sigma +1}x+c,\dots )(a,b,c\in K)$$ (we leave to the reader the task of completing the definition expressing that (5.3) is preserved) $$h_k:(x,y,z,u,v,w)⟼(kx,k^{\sigma +1}y,k^{\sigma +2}z,k^{\sigma +3}u,k^{2\sigma +3}v,k^{2\sigma +4}w)(k\in K^{*})$$ $$\omega :(x_{*},x_i,x_{i\prime })⟼(x_{*},{\epsilon }_i{x}_{i\prime },{\epsilon }_i{x}_i)({\epsilon }_0={\epsilon }_1=-{\epsilon }_2=1),$$ and one has the

$G^{*}$ is doubly transitive on $\mathrm{\Gamma }\text{.}$ If $K$ has more than three elements, it is generate by the $t_{a,b,c}$ $\text{(}a,b,c\in K\text{),}$ $h_k$ $\text{(}k\in K^{*}\text{)}$ and $\omega \text{.}$ Every element of $G^{*}$ may be put, in one and only one fashion, in one of the two forms $t_{a,b,c}·h_k$ or $t_{a,b,c}·h_k·\omega ·t_{a_1,b_1,c_1}\text{.}$

$G^{*}\left(f\right)=G^{*}\text{.}$

Remarks.

1.	The determinants of $t_{a,b,c},$ $h_k$ and $\omega$ are 7th powers, so $G^{*}$ is a subgroup of ${PSL}_7\left(K\right),$ which is anyway already true (and well known) for $G(E,L)\text{.}$
2.	There is no doubt that, just as $B_2^{*}$ and $G_2^{*}\text{,}$ $F_4^{*}$ may be defined as the group of elements of $F_4$ (of Tôhoku) which "preserve" a "polarity" of the geometry associated to it (cf. [10], notably paragraph 11), and that the elements of the geometry incident to their "polar" form the generalised octagon in question in paragraph 2.

A class of doubly transitive groups

Let $G$ be a doubly transitive group of permutations of a set $\mathrm{\Gamma }$ such that the subgroup leaving fixed a point $\infty$ of $\mathrm{\Gamma }$ possesses an invariant subgroup $U_{+}\text{,}$ simply transitive on $\mathrm{\Gamma }-\left\{\infty \right\},$ $0$ and $e$ two points of $\mathrm{\Gamma }-\left\{\infty \right\},$ $E$ the group of elements of $G$ preserving $\infty$ and $0,$ $U_{-}$ the conjugate of $U_{+}$ which has $0$ as a fixed point, $G_0$ the group generated by $U_{+}$ and $U_{-}\text{,}$ $H_0=H\cap G_0,$ $t$ the element of $U_{+}$ which moves $e$ to $0,$ $\bar{t}$ the element of $U_{-}$ which moves $\infty$ to $e\text{,}$ $t\prime$ the element of $U_{+}$ which moves $0$ to ${\bar{t}}^{-1}\left(\infty \right),$ and $\omega$ the transformation $t·\bar{t}·t\prime \in G_0,$ which exchanges $0$ and $\infty \text{.}$

The group $G_0$ is manifestly transitive on $\mathrm{\Gamma }\text{;}$ it thus contains all the conjugates of $U_{+}$ in $G\text{.}$ It is thus the smallest invariant subgroup of $G$ containing $U_{+}\text{.}$

$U_{+}$ being simply transitive on the set $\mathrm{\Gamma }-\left\{\infty \right\},$ one may endow it with the structure of the "abstract" group $U$ (which will be denoted additively) having $0$ for the identity element and such that $U=\{t_a,a\in U\}$ with $$\begin{array}{cc}{\displaystyle t_a:x⟼a+x(x\in \mathrm{\Gamma })}& \text{(6.1)}\end{array}$$ (we put $a+\infty =\infty \text{).}$ The elements of $H$ are automorphisms of $U\text{;}$ the action of $H$ on $U$ will be denoted multiplicatively (i.e. we will put $hx=h\left(x\right)\text{).}$ We easily see that $G$ (resp. $G_0\text{)}$ is composed of transformations $$\begin{array}{cc}{\displaystyle t_a·h:x⟼a+hx(a\in U)}& \text{(6.2)}\end{array}$$ $$\begin{array}{cc}{\displaystyle t_a·\omega ·t_b·h:x⟼a+\omega (b+hx)(a,b\in U),}& \text{(6.3)}\end{array}$$ with $h\in H$ (resp. $H_0\text{).}$ The transformation ${\omega }^2=\epsilon$ belongs to $H_0\text{,}$ and thus one may write $$\begin{array}{cc}{\displaystyle \omega \left(\omega \left(x\right)\right)=\epsilon x(\epsilon \in H_0)\text{.}}& \text{(6.4)}\end{array}$$ Recalling the definition of $t\text{,}$ $\bar{t}$ and $t\prime ,$ we see that $t=t_{-e},$ $\bar{t}={\omega }^{-1}·t_{\omega \left(e\right)}·\omega$ and $t\prime =t_{e\prime },$ with $e\prime ={\omega }^{-1}(-\omega \left(e\right))\text{.}$ Taking account of $t·\bar{t}·t\prime =\omega ,$ one then finds the relation $\omega ·t_e·\omega =t_{\omega \left(e\right)}·\omega ·t_{e\prime }\text{.}$ Generally, we consider, for every $a\in U\text{,}$ the transformations $\omega ·t_a·\omega$ and $t_{\omega \left(a\right)}·\omega ·t_{a\prime },$ with $a\prime ={\omega }^{-1}(-\omega \left(a\right))\text{;}$ these both move $\infty$ to $\omega \left(a\right)$ and $0$ to $0,$ and thus their quotient is an element ${\rho }_a$ of $H_0\text{,}$ which one may write $$\begin{array}{cc}{\displaystyle \omega (a+\omega \left(x\right))=\omega \left(a\right)+\omega ({\omega }^{-1}(-\omega \left(a\right))+{\rho }_{a}x)({\rho }_a\in H_0)}& \text{(6.6)}\end{array}$$ in particular $$\begin{array}{cc}{\displaystyle {\rho }_a=1\text{.}}& \text{(6.7)}\end{array}$$ We also note that putting $x={\omega }^{-1}\left(a\right)$ in (6.6), it becomes $$\begin{array}{cc}{\displaystyle {\rho }_a·{\omega }^{-1}\left(a\right)={\omega }^{-1}(-\omega \left(a\right))\text{.}}& \text{(6.8)}\end{array}$$

Let $H_1$ be the group generated by the ${\rho }_a\text{,}$ the ${\bar{\rho }}_a$ and $\epsilon \text{.}$ It is clear that $H_1\subseteq H_0\text{.}$ On the other had, we check easily that the transformations (6.2) and (6.3), with $h\in H_1,$ form a group which evidently contains $U_{+}$ and $U_{-}\text{,}$ and thus $G_0\text{.}$ It follows that $H_0=H_1\text{.}$

An easy computation shows that the preceding results admit a converse. More precisely,

Let $U$ be a group denoted additively, $H$ a group of automorphisms of $U$ and $\omega$ a permutation of $U-\left\{0\right\}\text{.}$ We assume that there exists a pemutation $h\mapsto \bar{h},$ a map $\rho :U-\left\{0\right\}\to H$ and elements $\epsilon \in H$ and $e\in U-\left\{0\right\},$ such that one has the identities (6.4), (6.5), (6.6) and (6.7) for all $h\in H$ and $a,x\in U-\left\{0\right\}$ $\text{(}a\ne -\omega \left(x\right)\text{),}$ and let $H_0$ is the subgroup generated by the ${\rho }_a\text{,}$ the ${\bar{\rho }}_a$ $\text{(}a\in U-\left\{0\right\}\text{)}$ and $\epsilon \text{.}$ We also denote by $\mathrm{\Gamma }=U\cup \left\{\infty \right\}$ the set obtained by adjoining a point $\infty$ to $U\text{,}$ and we put $a+\infty =h\infty =\omega \left(0\right)=\infty$ $\text{(}a\in U,h\in H\text{),}$ and $\omega \left(\infty \right)=0\text{.}$ Then, the transformations of $\mathrm{\Gamma }$ defined by (6.2) and (6.3), with $h\in H$ (resp. $H_0\text{)}$ form a doubly transitive group. The transformations (6.1) form an invariant subgroup $U_{+}$ of the group of elements of $G$ which preserve $\infty \text{;}$ it is generated by any two conjugates of $U_{+}$ in $G\text{.}$ $H_0$ is an invariant subgroup of $H$ and $G/G_0\cong H/H_0\text{.}$ 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.

Examples.

1.	Let $K$ be a field, $U=K^{+}$ (additive group), $H=K^{*}$ (acting in the obvious way on $H\text{)}$ and $\omega \left(k\right)=-k^{-1},$ whence $\bar{k}=k^{-1},$ ${\rho }_k=k^2$ and $H_0=K^2\text{.}$ Then $G={PGL}_2\left(K\right)$ and $G_0={PSL}_2\left(K\right)\text{.}$
2.	Let $K$ be a field of characteristic 2 where the Frobenius possesses a square root $\sigma \text{,}$ $U=\left\{(x,y)\hspace{0.17em}\right|\hspace{0.17em}x,y\in K\}$ with the group law $$(x+y)+(x\prime +y\prime )=(x+x\prime ,y+y\prime +x^{\sigma }x\prime ),$$ $H=K^{*},$ acting on $U$ in the following way: $$k·(x,y)=(kx,k^{\sigma +1}y),$$ and, putting $(x,y)=a,$ $\omega \left(a\right)=(\frac{y}{z\left(a\right)},\frac{x}{z\left(a\right)})$ with $z\left(a\right)$ defined by (4.2.1), whence $\bar{k}=k^{-1},$ ${\rho }_a={\left(z\left(a\right)\right)}^{2-\sigma }$ $\left({}^9\right)$ and $H_0={K^{*}}^{2-\sigma }={K^{*}}^{\sigma }\text{.}$ Then $G$ and $G_0$ are the group $G^{*}$ $\text{(}=B_2^{*}\text{)}$ and $G_0^{*}$ of paragraph 4.
3.	Let $K$ be a field of characteristic 3 where the Frobenius possesses a square root $\sigma \text{,}$ $U=\left\{(x,y,z)\hspace{0.17em}\right|\hspace{0.17em}x,y,z\in K\}$ with the group law $$(x,y,z)+(x\prime ,y\prime ,z\prime )=(x+x\prime ,y+y\prime +x^{\sigma }x\prime ,z+z\prime -xy\prime +yx\prime -x^{\sigma +1}x\prime ),$$ $H=K^{*},$ acting on $U$ in the following way, $$k·(x,y,z)=(kx,k^{\sigma +1}y,k^{\sigma +2}z),$$ and, putting $a=(x,y,z),$ $\omega \left(a\right)=(-\frac{v\left(a\right)}{w\left(a\right)},-\frac{u\left(a\right)}{w\left(a\right)},-\frac{z}{w\left(a\right)}),$ with $u\left(a\right)\text{,}$ $v\left(a\right)$ and $w\left(a\right)$ defined by (5.3), whence $\bar{k}=k^{-1},$ ${\rho }_a={w\left(a\right)}^{2-\sigma }$ $\left({}^{10}\right),$ $H_0=M^{2-\sigma },$ where $M$ denotes the subgroup of $K^{*}$ generated by all the $w\left(a\right)\text{.}$ Then $G$ is the group $G^{*}$ $\text{(}=G_2^{*}\text{)}$ of paragraph 5. Noting that $2-\sigma$ is an automorphism of $K^{*}$ (since $(2-\sigma )(2+\sigma )=1\text{),}$ one sees that $G/G_0\cong H/H_0\cong K^{*}/M\text{.}$

Remark. The theorem 7 and the preceding furnishes a completely elementary definition of the groups $B_2^{*}$ and $G_2^{*}$ 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 $G_0$ contains $U_{+}\text{,}$ it coincides with $G_0\text{.}$

This results from the fact that, $G_0$ is transitive on $\mathrm{\Gamma }\text{,}$ the conjugates of $U_{+}$ on $G_0$ are all the conjugates of $U_{+}$ in $G\text{.}$

If $H$ is commutative, $H_0$ contains all the elements of the form $\bar{h}·h^{-1}\text{.}$

In effect, since $G/G_0\cong H/H_0$ is commutative, $G_0$ contains the derived group of $G\text{,}$ so $\bar{h}{h}^{-1}$ is none other than the commutator of $\omega$ and $h\text{.}$

If $H_0$ is commutative and if there exists at least one element $h\in H_0$ such that the equation $x-\bar{h}·h^{-1}·x=a$ has a solution for all $a\in U\text{,}$ then $G_0$ is simple. If all $H$ is commutative, $G_0$ is the derived group of $G\text{.}$

		Proof.
	Let $J$ be a nontrivial invariant subgroup of $G\text{.}$ This will be doubly transitive, $J$ contains at least one transformation taking $0$ to $\infty ,$ let $x\mapsto a+h_1·\omega \left(x\right)$ $\text{(}{h}_1\in H_0\text{).}$ If $H_0$ is commutative, the commutator of $x\mapsto hx$ $\text{(}H\in H_0\text{)}$ and of this transformation is of the form $x\mapsto b+\bar{h}·h^{-1}·x\text{.}$ If $h$ satisfies the condition of the statement, this last transformation possesse at least one fixed point outside of $\infty ,$ and it is moved to $\bar{h}·h^{-1}:x\mapsto \bar{h}·h^{-1}·x$ by every transformation of $G_0$ which preserves $\infty$ and which brings the fixed point to $0\text{.}$ Finally, the commutator of $\bar{h}·h^{-1}$ and of $t_c$ (cf. (6.1)) is $t_d\text{,}$ with $d=c-\bar{h}·h^{-1}·c\text{.}$ However, due to the hypothesis made on $h\text{,}$ when $c$ is from $U\text{,}$ it is the same as $d\text{.}$ The second part of the statement follows from the simplicity of $G_0$ and the relation $G/G_0\cong H/H_0\text{.}$ $\square$

Theorem 8 has as an immediate consequence the part of Theorem 1 which concerns $B_2^{*}$ and $G_2^{*}\text{.}$ We also note the following corollary.

(Suzuki) Let $G$ be a doubly transitive group of permutations of a finite set $\mathrm{\Gamma }$ of $2m+1$ points. If $G$ contains permutations different from the identity having two fixed points, and if, conversely, the identity is the only permutation in $G$ which has more than two fixed points, then $G$ is simple.

		Proof.
	The subgroup $G_{\infty }$ of $G$ formed by the elements of the group that leave fixed a given point $\infty \in \mathrm{\Gamma }$ is a Frobenius group, so that the elements of $G_{\infty }$ which do not leave fixed any point other than $\infty$ constitutes an invariant subgroup $U_{+}$ of $G_{\infty },$ simply transitive on $\mathrm{\Gamma }-\left\{\infty \right\}\text{.}$ One may thus apply the preceding considerations. A transformation in $G$ which exchanges $0$ and $\infty$ may not have an odd cycle $>1,$ 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 $\omega$ and $H$ be defined as above, $h_1$ and $h_2$ two elements of $H$ and ${\omega }_i=h_i\omega$ $\text{(}i=1,2\text{).}$ The transformations ${\omega }_1,$ ${\omega }_2,$ $\omega ·{\omega }_1\cdots {\omega }_2$ and ${\omega }_1·\omega ·{\omega }_2,$ which all exchange $0$ and $\infty ,$ are involutions. It follows that $$h_1{h}_2{\omega }_1·\omega ·{\omega }_2·\omega ={({\omega }_1·\omega ·{\omega }_2)}^{-1}\omega ={\omega }_2·\omega ·{\omega }_1·\omega =h_2{h}_1,$$ giving that $H$ is commutative, and $${\bar{h}}_1=\omega ·h_1·\omega =\omega ·{\omega }_1·\omega ·\omega =\omega ·{\omega }_1=h_1^{-1}\text{.}$$ The order of $H$ divides $2m-1$ (since, by hypothesis, $H$ is simply transitive on the different orbits of $\left\{0\right\}$ and $\left\{\infty \right\}\text{);}$ it is thus odd. It is a result that each element of $H$ is a square, and is thus of the for $\bar{h}·h^{-1},$ and one has $G=G_0,$ in virtue of the proposition 4. On the other hand, if there are $x,x\prime \in \mathrm{\Gamma }-\left\{\infty \right\}$ and $h\in H$ $\text{(}h\ne 1\text{),}$ the equality $x-hx=x\prime =hx\prime$ leads to $-x\prime +x=h·(-x\prime +x),$ whence $x=x\prime \text{;}$ in other words, the map of $\mathrm{\Gamma }-\left\{\infty \right\}$ to itself defined by $x\mapsto x-hx$ is injective, thus bijective; and one is in the conditions for application of theorem 8. The corollary is thus proved. $\square$

Remark. The group $B_2^{*}$ over a finite field $K$ satisfies the conditions of the preceding statement.

Footnotes

$\left({}^1\right)$ This paragraph and the two following are directly inspired by two still unpublished articles of R. REE [4] and [5], who has kindly allowed us to make use of his manuscripts.

$\left({}^2\right)$ The automorphisms of $K$ are denoted exponentially; moreover, all the maps (and in particular the automorphisms of $G\text{)}$ act on the left.

$\left({}^3\right)$ 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.

$\left({}^4\right)$ It is not impossible that $M=K^{*}$ (whence $G_2=G_2^{*}\prime \text{)}$ for some $K\text{.}$

$\left({}^5\right)$ The following properties are not stated by R. REE, but they are deduced very easily from his results.

$\left({}^6\right)$ One may also check by direct computation that $\omega \in G^{*}\text{.}$ The only point is that a bit delicate in the proof consists in establishing that, on $\mathrm{\Gamma }\text{,}$ $z=0$ implies $x=y=0,$ but this results immediately from the relation $y^{\sigma +1}=x{z}^{\sigma }+z(x^{\sigma +1}+y),$ a consequence of 4.2.1.

$\left({}^7\right)$ Remark communicated to the author by Ms FELLEGARA.

$\left({}^8\right)$ This is essentially the definition of the group $B_2^{*}$ of SUZUKI.

$\left({}^9\right)$ ${\rho }_a$ is easily calculated by equating $z$ with the two sides of (6.8) and making use of the following, easy to establish, identities: $z(-a)=z\left(a\right),$ $z\left(\omega \left(a\right)\right)=z{\left(a\right)}^{-1},$ $z\left(ka\right)=k^{\sigma +2}z\left(a\right)\text{.}$

$\left({}^{10}\right)$ We equate $z$ with the two sides of (6.8) and we make use of the identities: $z(-a)=-z\left(a\right),$ $w(-a)=w\left(a\right),$ $z\left(\omega \left(a\right)\right)=-z\left(a\right)·\omega {\left(a\right)}^{-1},$ $w\left(\omega \left(a\right)\right)=w{\left(a\right)}^{-1},$ $z\left(ka\right)=k^{\sigma +2}z\left(a\right)\text{.}$

References

[1] BARLOTTI (Adriano). – Un'estensione del teorema di Segre-Kustanheimo, Boll. Unione mat. Italiana, v. 10, 1955, p. 498-506.

[2] BOURBAKI (Nicolas). – Algèbre, chap. 7. – Paris, Hermanna, 1948 (Act. scient. et ind., 1044; Eléments de Mathématique, 14).

[3] CHEVALLEY (Claude). – Sur certains groups simples, Tôhoku math. J., Series 2, v. 7, 1955, p. 14-66.

[4] REE (Rimhak). – A family of simple groups associated with the simple Lie algebra of type $\text{(}{G}_2\text{),}$ Amer. J. of Math. (to appear).

[5] REE (Rimhak). – A family of simple groups associated with the simple Lie algebra of type $\text{(}{F}_4\text{),}$ Amer. J. of Math. (to appear).

[6] SEGRE (Beniamino). – Le geometrie de Galois, Annali di Mathematica, Serie 4, v. 48, 1959, p. 1-96.

[7] SEGRE (Beniamino). – On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two, Acta Arithmetica, v. 5, 1959, p. 315-332.

[8] STEINBERG (Robert). – Variations on a theme of Chevalley, Pacific J. of Math., v. 9, 1959, p. 875-891.

[9] SUZUKI (Michio). – A new type of simple groups of finite order, Proc. Nat. Acad. Sc. U.S.A., v. 46, 1960, p. 868-870.

[10] 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).

[11] TITS (Jacques). – Let "formes réles" des group te type $E_6\text{,}$ Séminaire Bourbaki, v. 10, 1957/58, no. 162, 15 p.

[12] 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).

Notes and References

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.

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