Advanced Studies in Pure Mathematics 32, 2001 Groups and Combinatorics{in memory of Michio Suzuki pp. 1-39 Michio Suzuki Koichiro Harada \S 1 Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 \S 2 The Early Work of Michio Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 \S 3 Theory of Exceptional Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 \S 4 The $CA$-paper of Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 \S 5 Zassenhaus Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 \S 6 Suzuki’s Simple Groups $Sz(2^{n})\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 13$ \S 7 $ZT$-groups and Related Classi�cation Theorems . . . . . . . . . . . . . 17 \S 8 Group Theory in Japan before Suzuki. . . . . . . . . . . . . . . . . . . . . . . 29 \S 9 Michio Suzuki, my teacher and my mentor. . . . . . . . . . . . . . . . . . . 31 \S 1. Biographical Sketch 1926, October 2. Born in Chiba, Japan. 1942, April. Entered the Third High School of Japan located at Kyoto (Noboru Ito, Katsumi Nomizu, Hidehiko Yamabe were his seniors by one year and Singo Murakami was in the same class). 1945, April. Entered the University of Tokyo. Majored in mathematics. (Gaishi Takeuchi, Nagayoshi Iwahori, Tsuneo Tamagawa were friends of this period.) 1948, April. Entered the Graduate School of Tokyo University. Suzuki’s supervisor was Shokichi Iyanaga. Kenkichi Iwasawa had a profound in�uence on Suzuki. 1948-,51. Received a special graduate fellowship from the Government of Japan. 1951, April to ’52, January. Held a lecturership at Tokyo University of Education 1952, January to ’52, May. Held a graduate fellowship at University of Illinois at Urbana-Champaign. Received April 27, 1999. Revised May 17, 1999. K. Harada 1952, May. Received the Doctor of Science Degree from the University of Tokyoin absentia. 1952. Spent two months in the summer at University of Michigan. R. Brauer was a professor of Mathematics at Michigan. J. Walter, W. Feit were graduate students there. 1952, September to ’53, May. Held a post-doctoral fellowship at Univer- sity of Illinois at Urbana-Champaign. 1952, November. Married to a daughter Naoko of Yasuo Akizuki (then Professor at Kyoto University). 1953, September to ’55, May. Held a research associateship at University of Illinois. 1955, September. Promoted to an assistant professor at University of Illinois. 1956, September to ’57, May. Held a research associateship at Harvard University. 1958, September. Promoted to an associate professor at University of Illinois. 1959, September. Promoted to a full professor at University of Illinois. 1960, Discovered a new series of �nite simple groups $Sz(q)$ . 1960-,61. Held a visiting appointment at the University of Chicago. 1962. Invited to speak at the International Congress of Mathematicians in Stockholm. 1962-,63. Held a Guggenheim Fellowship. 1962, September to ’63, May. Held a membership at the Institute for Advanced Study, Princeton. 1967. Discovered a sporadic simple group Suzuki of order 448,345,497,600. 1968-,69. Held a visiting appointment at the Institute for Advanced Study, Princeton, $NJ$ . 1970. Invited to speak at the International Congress of Mathematicians in Nice, France. 1974. Received the Academy Prize from the Japan Academy. 1987. The conference of group theory and combinatorics for the occasion of Suzuki’s 60th birthday was held in Kyoto, Japan. 1991. Awarded an honorary doctoral degree from the University of Kiel, Germany. 1997. The conference of group theory and combinatorics for the occasion of Suzuki’s 70th birthday was held in Tokyo, Japan. Michio Suzuki 3 1998, May 31. Died at the age of 71. (Evariste Galois died on May 31, 1832.) (A cancer was discovered in his liver early in February, 1998. Left for Japan. Received the same diagnosis. Stayed in the hospital (February 12 - March 13), at a Guest House of the International Christian University (March 14- April 17). Back to hospital on April 18.) June 7: Funeral Service at the International Christian University, Mitaka, Japan. September 18: Memorial Service at the University of Illinois, Ur- bana, Illinois. \S 2. The early work of Michio Suzuki Among Suzuki’s earliest research papers are: [2] On the �nite group with a complete partition, 1950. [5] A characterization of simple groups $LF(2,p)$ , 1951. [6] On �nite groups with cyclic Sylow subgroups for all odd primes, 1955. In [2], Suzuki investigates the structure of a �nite group $G$ having a partition by its subgroups $H_{i}$ , $i=1$ , $\ldots$ , $n$ : $n$ , $H_{i}\cap H_{j}=1$ if $i\neq j$ . $G=i=1\cup H_{i}$ A partition of $G$ is called complete if $H_{i}$ is cyclic for all $i=1$ , $\ldots$ , $n$ . The research on groups having a complete partition goes back at least to P. Kontorovich [Sur la representation d’un groupe �ni sous la forme d’une somme directe de sous-groupes, I. Rec. Math. (Mat. Sbornik), 5 (47) (1939), 283-296]. In [2] Suzuki considers groups having a complete partition. Exam- ples of such groups are $PGL(2, q)$ and $PSL(2, q)$ where $q$ is a power of a prime. In [2], however, Suzuki determines only nonsimple groups having a complete partition. It is shown �rst that if $G$ is a nonsimple, nonsolvable �nite group with a complete partition then a minimal nor- mal subgroup $N$ of $G$ is of index 2. The proof proceeds by induction on the order of $G$ , since the complete partitionability carries over to its subgroups and even to its factor groups as Suzuki shows. Suzuki next shows that the Sylow 2-subgroups of $G$ are dihedral, and that for any odd prime $p$ , any two distinct Sylow $p$ subgroups of $G$ have a trivial intersection. He then uses a counting argument to obtain a con�guration in which the group $G$ is a sharply triply transitive per- mutation group acting on the coset space $G/M$ where $M$ is a suitable 4 K. Harada subgroup of $G$ obtained in the counting argument mentioned above. Therefore Suzuki is able to use the result of Zassenhaus [Kennzeich- nung.endlicher linearer Gruppen als Permutationsgruppe, Hamb. Abh., 11(1936), 17-40], who had classi�ed, among other results, all such per- mutation groups, hence the theorem: Theorem. Let $G$ be a nonsimple, nonsolvable ��nite group with $a$ complete partition. Then $G$ is isomorphic to the full linear fractional group $PGL(2, q)$ where $q$ is a power of an odd prime. Character theory is not used in [2]. This paper shows that Suzuki was a young mathematician of foresight. He was able to recognize the importance of the groups $PSL(2, q)$ and Zassenhaus’ work. The concept of a group having a partition does not appear to be very important on its own right, but it should be mentioned that the in�nite series of new simple groups $Sz(q)$ discovered by Suzuki in 1960 does have a partition, though not a complete partition. Suzuki completes the classi�cation of all (semi) simple groups with a partition in 1961 [18]. As for the paper [5], let us �rst observe that the subgroups of the simple groups $PSL(2,p)$ for a prime $p$ are of the types: (1) metacyclic groups; (2) the alternating group $A_{4}$ of degree 4; (3) the symmetric group $S_{4}$ of degree 4; or (4) the alternating group $A_{5}$ of degree 5. In [5], Suzuki characterizes $PSL(2,p)$ by this property. Let $G$ be a �nite simple group such that all of its subgroups are of types $(1)-(4)$ mentioned above. Suzuki �rst shows that $G$ possesses a complete partition in the sense of the paper [2]. Among all papers of Suzuki, the theory of exceptional characters �rst appeared here in [5]. Using this theory and Brauer’s work on a group whose order is divisible by a prime to the �rst power, Suzuki was able to show that $G$ possesses an irreducible character of degree $\frac{1}{2}(p\pm 1)$ for some prime $p$ . He next applies a result of $H.F$ . Tuan [On groups whose orders contains a prime number to the �rst power, Ann. of Math., 45(1944), 110-140] to complete the characterization of $PSL(2, p)$ . As he recognized the importance of studying the simple groups $PSL(2, q)$ , he began doing research on them from various points of view : in [2] as groups having a partition, in [5] as groups having only a special set of isomorphism classes of subgroups, etc. Although the papers [2] or [5] of Suzuki might perhaps not be among his better works, if they are considered as stand-alone papers, the line of research in this direction served him well and it culminated in the Michio Suzuki 5 discovery of the simple groups $Sz(q)$ and the classi�cation of all Zassen- haus groups (which was completed by a joint effort of Zassenhaus, Feit, Ito and Suzuki). The paper [6] is also part of Suzuki’s continuing efforts to under- stand the simple groups $PSL(2,p)$ . Its content is fully explained in the title. Its introduction begins with ’The purpose of this paper is to de- termine the structure of some �nite groups in which all Sylow subgroups of odd order are cyclic.The assumption on Sylow subgroups simpli�es the structure of groups considerably, but the structure of 2-Sylow sub- groups might be too complicated to make any de�nite statement on the structure of the groups. In this paper, therefore, we shall make another assumption on 2-Sylow subgroups, $\cdots$ ’. In fact, he assumes that the Sylow 2-subgroups of $G$ are either (a) di- hedral or (b) generalized quaternion. The Sylow 2-subgroups of $SL(2,p)$ are, as is well known, generalized quaternion if $p$ is odd. Suzuki shows that the group $G$ contains a normal subgroup $G_{1}=Z\times L$ of index at most 2 such that $L\cong PSL(2,p)$ if (a) holds, and $L\cong SL(2,p)$ if (b) holds. Moreover, $Z$ is a group of odd order all of whose Sylow sub- groups are cyclic. Frobenius and Burnside treated groups such that all of their Sylow subgroups are cyclic and showed that all such groups are solvable, in fact all such groups are metacyclic. Zassenhaus classi�ed all solvable groups with the same assumption on Sylow subgroups for odd primes but with the weaker assumption for the prime 2 that a Sylow 2-subgroups has a cyclic subgroup of index 2. \S 3. Theory of exceptional characters ’Perhaps the �rst mathematician of the post war generation who mastered Brauer’s work in group theory was M. Suzuki. He came to the United States in the early �fties and he has made many signi�cant contributions to the theory of simple groups (from W. Feit [$R.D$ . Brauer, Bull (New Series). Amer. Math. Soc, 1 (1979), 1-20]) ’. Having begun his research on exceptional characters in [5], Suzuki wrote a couple of papers on the subject [13], [19], and several papers in which the theory played a crucial role [6], [8], [9], [10], [17]. In his work on the theory of modular representations, Brauer de- �ned the concept of an exceptional character. Brauer and Suzuki inde- pendently extended this concept of exceptional characters at about the same time, around 1950. Although the basic assumption of the theory can be loosened from the one given below, we will show it in the simplest but most important setting. 6 K. Harada We are typically interested in a �nite group $G$ having an abelian subgroup $A$ such that the centralizer of every nonidentity element of $A$ is contained in $A$ (hence it is equal to $A$ itself and $A$ is a maximal abelian subgroup of $G$ ). The simple group $PSL(2, q)$ contains a couple of conjugacy classes of such abelian subgroups. For Suzuki, a motivation to extend the theory of exceptional characters must have come from his investigation of the simple group $PSL(2, q)$ . Under this condition on $G$ and on $A$ , the following conditions hold: (1) $A$ is an abelian $TI$ subgroup of $G$ : i.e. $A\cap A^{g}=A$ or 1 for every element $g$ of $G$ . (2) The normalizer $N=N_{G}(A)$ of $A$ in $G$ is a Frobenius group. Let $l$ $=[N : A]$ and $w=\frac{|A|-1}{l}$ . Then $G$ possesses exactly $w$ conjugacy classes of elements represented by nonidentity elements of $A$ . The Frobenius group $N$ possesses $l$ irreducible characters of degree 1, all of which contain $A$ in their kernels. In addition to those linear characters, $N$ possesses $w$ irreducible characters not containing $A$ in $l$ their kernels, and all of them have degree . Those are all the irreducible characters of $N$ . Thus $N$ possesses exactly $l+w$ irreducible characters. We can actually obtain the irreducible characters of $N$ of degree $l$ as follows. Let $\{\psi_{i}, i=1, \ldots, w\}$ be the complete set of representa- tives of $N$-orbits (by conjugation) consisting of nonidentity irreducible characters of $A$ and $\Psi_{i}=\psi_{i}^{N}$ be the corresponding induced character of $\psi_{i}$ to $N$ . By computing the inner product directly, we see that $\Psi_{i}$ is an irreducible character of $N$ for all $i$ . We thus obtain $w$ irreducible characters of $N$ of degree $l$ . The remaining irreducible characters of $N$ (of degree 1) will appear as constituents of the induced character of the trivial character of $A$ . Let $\Psi_{i}^{G}$ , $i=1$ , $\ldots$ , $w$ be the corresponding induced characters to $G$ . We compute that $\Psi_{i}^{G}(g)=0$ if $g$ is not conjugate to an element of $A\backslash 1$ and $\Psi_{i}^{G}(g)=\Psi_{i}(a)$ if $g$ is conjugate to an element $a$ of $A\backslash 1$ . Thus $\langle\Psi_{i}^{G}, \Psi_{i}^{G}\rangle_{G}-(\Psi_{i}^{G}(1))^{2}=\langle\Psi_{i}, \Psi_{i}\rangle_{N}-(\Psi_{i}(1))^{2}$ . Therefore, the norm $||\Psi_{i}^{G}||c$ is almost determined by the norm $||\Psi_{i}||_{N}$ , but not completely so since $\Psi_{i}^{G}(1)$ is an unknown number. If we can �nd a way to eliminate the ambiguity then it will be nice. Now assume, in addition to (1) and (2) mentioned above: (3) $w\geq 2$ . Consider the generalized character $\Psi_{i}-\Psi_{j}$ , $i\neq j$ , of $N$ . Then we obtain $||\Psi_{i}^{G}-\Psi_{j}^{G}||=2$ Michio Suzuki 7 since $||\Psi_{i}^{G}-\Psi_{j}^{G}||_{G}=||\Psi_{\dot{0}}-\Psi_{j}||_{N}=2$ holds. Therefore, $\Psi_{i}^{G}-\Psi_{j}^{G}=$ $\epsilon_{ij}(\ominus_{i}.-\Theta_{j})$ where $\Theta_{i}$ , $\Theta_{j}$ are irreducible characters of $G$ and $\epsilon_{ij}=\pm 1$ . Actually $\epsilon_{ij}$ is independent of $i,j$ and so $\Psi_{i}^{G}-\Psi_{j}^{G}=\epsilon(\ominus_{i}-\Theta_{j})$ , $\epsilon=\pm 1$ . This implies that $\Psi_{i}^{G}=\epsilon\Theta_{i}+\triangle$ where $\triangle$ is a generalized character of $G$ independent of $i=1$ , $\ldots$ , $w$ . The irreducible $characters\ominus_{i}$ , $i=1$ , $\ldots$ , $w$ obtained above are called exceptional characters of $G$ associated with A. (W. Feit was able to extend the exceptional character theory by dropping the condition that $A$ is abelian. Feit still needed that $A$ is nilpotent and is not isomorphic to a certain type of $p$-group. A further extension was obtained by D. Sibley.) Exceptional characters satisfy the following properties. Let $D$ be the set of all elements of $G$ not conjugate to any element of $A\backslash 1$ . (I) $\Theta_{i}(\sigma)=\Theta_{j}(\sigma)$ if $\sigma\in D$ for every pair $i$ , $j$ . In particular all excep- tional characters $\Theta_{i}$ have the same degree. (II) The exceptional characters are linearly independent on the conju- gacy classes $\{C_{1}, \ldots, C_{w}\}$ of $G$ represented by the elements of $A\backslash 1$ : i.e. if $\sum_{i=1}^{w}a_{i}\Theta_{i}(\sigma)=0$ for all $\sigma\in\bigcup_{i=1}^{w}C_{i}$ , then $a_{i}=0$ for all $i=1$ , $\ldots$ , $w$ . (III) If $B$ is another abelian subgroup of $G$ not conjugate to $A$ but satisfying the same property as $A$ does, then the exceptional characters for $A$ are nonexceptional characters for $B$ . Therefore if $G$ has many nonconjugate abelian subgroups of the same property, then the majority of the irreducible characters of $G$ will be exceptional characters associated with some abelian subgroup $A$ . Using those irreducible characters, one can obtain strong numerical conditions on the order of $G$ . \S 4. The $CA$-paper of Suzuki Theorem ([8]). Let $G$ be $a$ ��nite simple group such that the cen- tralizer of every nonidentity element is abelian. Then the order of $G$ is even. Let us quote Thompson �rst: $?$ ‘A third strategy (or was it a tactic ) in OOP (Odd Order Paper) attempted to build a bridge from Sylow theory to character theory. The far shore was marked by the granite of Suzuki’s theorem on $CA$-groups, 8 K. Harada �anked by W. Feit, M. Hall, Jr. and $J.G$ . Thompson [Finite groups in which the centralizer of any non-identity element is nilpotent, Math. Z., 74(1960), 1-17]. The bridge was built of tamely embedded subsets with their supporting subgroups and associated $tau(\tau)$ isometry. The near $E$ shore was dotted with the -theorems and the uniqueness theorems. Suzuki’s $CA$-theorem is marvel of cunning. In order to have a gen- uinely satisfying proof of the odd order theorem, it is necessary, it seems to me, not to assume this theorem. Once one accepts this theorem as a step in a general proof, one seems irresistibly drawn along the path which was followed. To my colleagues who have grumbled about the tortuous proofs in the classi�cation of simple groups, I have a ready answer: �nd another proof of Suzuki’s theorem (from J.G. Thompson [Finite Non-Solvable Groups, in Group Theory:essays for Philip Hall, ’ Academic Press, (1984), 1-12]) Now let $G$ be a �nite group such that the centralizer of every non- identity element is abelian. Let us call such a group $G$ a $CA$ group. Already in $1920’ s$ , it was known that every $CA$-group is either solvable or simple (L. Weisner [Groups in which the normalizer of every element except the identity is abelian, Bull. Amer. Math. Soc, 31(1925), 413- 416]). So let us assume that our $CA$ group $G$ is nonabelian and simple. Let $g$ be a nonidentity element of $G$ . Then the centralizer $A=C_{G}(g)$ is a proper abelian subgroup of $G$ . Let $1\neq h\in A$ . Then $C_{G}(h)\supset A$ . The fact that $C_{G}(h)$ is abelian forces the equality $C_{G}(h)=A$ , thus $A$ is a maximal abelian subgroup of $G$ , and $A$ is a $TI$-set. The rudiments of group theory also show that $A$ is a Hall subgroup of $G$ , i.e. $gcd(|G : A|, |A|)=1$ . If the normalizer $N=N_{G}(A)$ is equal to $A$ itself, then $N_{G}(P)=C_{G}(P)$ for a Sylow $p$ subgroup $P$ of $A$ for some prime $p$ . Since $P$ is a Sylow $p$-subgroup of $G$ also, Burnside’s theorem implies that $G$ is nonsimple. Thus $N>A$ and $N$ is a Frobenius group. In order to apply the exceptional character theory effectively, we need one more condition : $w\geq 2$ where $w=\frac{|A|-1}{l}$ , $l$ $=[N : A]$ . For this purpose, we henceforth assume that $G$ is of odd order as this is the case Suzuki treats. Then $|A|$ and $l$ are both odd, and so $w$ can not be equal to 1. Hence $w\geq 2$ as desired. Let $\{A_{i}, i=1, \ldots, n\}$ be a complete set of representatives of the conjugacy classes of maximal abelian subgroups of $G$ and we put $\prime N_{i}=$ $Nc(Ai)$ . We have shown that $N_{i}>A_{i}$ and $N_{i}$ is a Frobenius group for all $i$ . Moreover, every element of $G\backslash 1$ has a representative in $\bigcup_{i=1}^{n}A_{i}$ . Michio Suzuki 9 Since each $A_{i}$ is a $TI$-set, we have $|G|=1+\sum_{i=1}^{n}[G:N_{i}](|A_{i}|-1)$ . Each $A_{i}$ gives rise to $w_{i}=+(|A_{i}|-1/l_{i})$ (where $l_{i}=[N_{i}$ : $A_{i}]$ ) exceptional characters and so $G$ has $\sum_{i=1}^{n}w_{i}exceptiona1$ characters in total. On the other hand, $G$ possesses precisely 1 $+\sum_{i=1}^{n}w_{i}$ conju- gacy classes. Therefore every nonidentity irreducible character of $G$ is exceptional for some $A_{i}$ . Suzuki puts all of this information together and starts a counting argument. In three pages, he is able to reach a contradiction. This $CA$-paper of Suzuki was received by the editors on December 24, 1954 but was published in 1957. Suzuki knew who was the referee. It was none other than R. Brauer. Apparently Brauer did not understand some argument of Suzuki and left it there for a (great) while. Suzuki submitted a revised version two years later and the paper was published soon. ’At the time its importance was not fully grasped, either by him or by others, as it seemed to be simply an elegant exercise in character theory. However, the result and the methods used had a profound impact on much succeeding work (W. Feit [Obituary written for Michio Suzuki, Notices of Amer. Math. Sci., Vol. 46(1999) $])$ . ’ L. Redei [Ein Satz \"uber die endlichen einfachen Gruppen, Acta. Math., 84(1950), 129-153] considered �nite simple groups such that ev- ery proper subgroup of every maximal subgroup is abelian. He showed that the alternating group of degree 5 is the only such group of even or- der. One obtains, as a corollary to the main theorem of this paper, that there is no such group of odd order. Moreover, Suzuki proved that the word abelian in Redei’s theorem can be replaced by nilpotent to assert the same conclusion. Suzuki uses the assumption that $G$ is of odd order only to assert $w\geq 2$ and so this method can go farther under a suitable assumption. In fact, R. Brauer, M. Suzuki, and $G.E$ . Wall, more or less independently proved: Theorem. If the centralizer of every element of $a$ ��nite group $G$ is abelian then either $G$ is solvable or $G$ is isomorphic to $PSL(2,2^{n})$ . In the published form of the Brauer-Suzuki-Wall Theorem [9], how- ever, it is stated as follows: 10 K. Harada Theorem. Let $G$ be a group of even order which satis��es the condition: (1) $I\acute{f}$ two cyclic subgroups $A$ and $B$ of even order of $G$ have a nontrivial intersection then there exists a cyclic subgroup $C$ of $G$ that contains both $A$ and $B$ . (2) $G=[G, G]$ . Then $G\cong PSL(2, q)$ for some prime power $q$ . One of my colleagues, Ronald Solomon, and I studied the latter theorem but could not conclude that it implies the former. We wrote a letter of inquiry to $G.E$ . Wall, who replied that they worked fairly independently with not a great deal of communication between them. He says also that the BSW paper (published version) was written by R. Brauer who did not have enough time to weld together three rather different versions and that the $CA$-groups of even order are not covered in any obvious way (in the published version), but they are covered in the ’behind scenes’ BSW versions. \S 5. Zassenhaus groups Let $V$ be a 2-dimensional vector space over a �eld $K$ and let $A=\left(\begin{aray}{l}\alpha & \beta\gama & \delta\end{aray}\right)$ $\in GL(V)$ be a $2\times 2$ matrix of nonzero determinant with entries in $K$ . The matrix $A$ acts on $V$ as a linear transformation and so the image of a line (1 dimensional subspace of $V$ ) is again a line. Since the structure of $GL(V)$ depends only on the dimension of $V$ and the �eld $K$ , we write $GL(2, K)$ for $GL(V)$ also. Let $P_{1}(K)$ be the set of all lines of V. $GL(2, K)$ acts on $P_{1}(K)$ . The scalar matrices $A=\left(\begin{aray}{l}\alpha & 0\0 & \alpha\end{aray}\right)$ are the only matrices that act trivially on $P_{1}(K)$ . Denote by $Z$ the set of all scalar matrices of $GL(2, K)$ . Then the factor group $PGL(2, K)=GL(2, K)/Z$ acts on $P_{1}(K)$ faithfully. If $\{u_{1}, u_{2}\}$ and $\{v_{1}, v_{2}\}$ are any pairs of linearly independent vec- tors of $V$ , then there is an element $g\in GL(2, K)$ such that $g(u_{1})=$ $v_{1}$ , $g(u_{2})=v_{2}$ . This implies that $PGL(2, K)$ is doubly transitive on $P_{1}(K)$ since if $[u]$ denotes the line spanned by the vector $u\in V$ , then $\overline{g}([u_{1}])=[v_{1}],\overline{g}([u_{2}])=[v_{2}]$ where $\overline{g}$ is the image of $g\in GL(2, K)$ in $PGL(2, K)$ . Put $SL(2, K)=\{g\in GL(2, K)|\det g=1\}$ and $PSL(2, K)=$ $SL(2, K)/Z\cap SL(2, K)$ . As is easily seen, $PSL(2, K)$ is also doubly