V.M. Sinitsin. On genetic codes of certain groups with 3-transpositions ... P. 184-188

Coxeter groups have numerous applications in mathematics and beyond, and B. Fischer's 3-transposition groups underly the internal geometric analysis in the theory of finite (simple) groups. The intersection of these classes of groups consists of finite Weyl groups $W(A_n)\simeq S_{n+1}$, $W(D_n)$, and $W(E_n)$ for $n=6,7,8$, simple finite-dimensional algebras, and Lie groups. In previous papers by A.I. Sozutov, A.A. Kuznetsov, and the author, systems $S$ of generating transvections (3-transpositions) of groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$ were found such that the graphs $\Gamma(S)$ are trees. A set $\{\Gamma_n\}$, $n\geq m$, of nested graphs is called an $E$-series if these graphs are trees, contain the subgraph $E_6$, and their subgraphs with vertices $m,m+1,\ldots,n$ are simple chains. In the present paper, we find genetic codes of the groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$, $8\leq 2m \leq 20$; these codes are close to the genetic codes of some Coxeter groups. Our main hypothesis is the following: the groups $Sp_{2m}(2)$ and $O^\pm_{2m}(2)$ (cases (ii)-(iii) in Fischer's theorem) can be obtained from the corresponding infinite Coxeter groups with the use of one or two additional relations of the form $w^2=1$. The graphs $I_n$ considered in this paper contain the subgraph $E_6$ and comprise an $E$-series of nested graphs $\{I_n\,\mid\,n=7, 8,\ldots\}$, in which the subgraph $I_n\setminus E_6$ is a simple chain. We prove that the isomorphisms $X(I_{4k+1})\simeq Sp_{4k}(2)\times Z_2$ and $X(I_{2m})\simeq O^\pm_{2m}(2)$ (the sign $\pm$ depends on $m$) hold for the groups $X(I_n)$ obtained from the Coxeter groups $G(I_n)$ by imposing an additional relation $(s_4^ts_7)^2=1$, where $t=s_3s_2s_1s_5s_6s_3s_2s_5s_3s_4$, if $n=4k +\delta$ ($\delta=0,1,2$). The proof uses the Todd-Coxeter algorithm from the GAP system.

Keywords: genetic codes, Coxeter groups and graphs, Weyl groups, 3-transposition groups, symplectic transvections

Received September 17, 2019

Revised October 25, 2019

Accepted November 18, 2019

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00566 A).

Vladimir Mihaylovich Sinitsin, Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: sinkoro@yandex.ru

REFERENCES

1.   Fischer B. Finite groups generated by 3-transpositions. Invent. Math., 1971, vol. 13, no. 3, pp. 232–246. doi: 10.1007/BF01404633 

2.   Aschbacher M. 3-transposition groups. Ser. Cambridge Tracts in Math., vol. 124, Cambridge: Cambridge University Press, 1997, 260 p. ISBN: 0-521-57196-0 .

3.   Gorenstein D. Finite simple groups. University Ser. in Math. N Y: Plenum Publishing Corp., 1982, 333 p. ISBN: 0-306-40779-5 . Translated to Russian under the title Konechnye prostye gruppy. Moscow: Mir Publ., 1985, 352 p.

4.   McLaughlin J. Some subgroups of $SL_n(F_2)$. Ill. J. Math., 1969, vol. 13, no. 1, pp. 108–115. doi: 10.1215/ijm/1256053741 

5.   Sozutov A.I. Groups of type $\Sigma _4$ generated by 3-transpositions. Sib. Math. J., 1992, vol. 33, no. 1, pp. 117–124. doi: 10.1007/BF00972943 

6.   Sozutov A.I. On lie algebras with monomial basis. Sib. Math. J., 1993, vol. 34, no. 5, pp. 959–971. doi: 10.1007/BF00971409 

7.   Sozutov A.I., Kuznetsov A.A., Sinitsin V.M. About systems of generators of some groups with 3-transpositions. Sib. Elektron. Mat. Izv., 2013, vol. 10, pp. 285–301 (in Russian). doi: 10.17377/semi.2013.10.022 

8.   Bourbaki N. Groupes et algebres de Lie (Chapt. IV–VI). Paris: Hermann, 1968, 282 p. doi: 10.1007/978-3-540-34491-9 . Translated to Russian under the title Gruppy i algebry Li (glavy IV–VI). Moscow: Mir Publ., 1972, 334 p.

9.   Coxeter H.S.M., Moser W.O.J. Generators and Relations for Discrete Groups. Berlin: Springer-Verlag, 1972, 164 p. doi: 10.1007/978-3-662-21946-1 . Translated to Russian under the title Porozhdajushhie jelementy i opredeljajushhie jelementy diskretnyh grupp. Moscow: Nauka Publ., 1980, 240 p.

10.   Kondratiev A.S. Gruppy i algebry Li [Lie groups and Lie algebras. Ekaterinburg, 2009, 310 p. ISBN: 978-5-7691-2111-1 .

11.   Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A. Atlas of finite groups. Oxford: Clarendon Press, 1985, 252 p. ISBN: 0198531990 .

12.   O’Meara O.T. Lectures on symplectic groups. Indiana: University of Notre Dame, 1976. Translated to Russian under the title Lekcii o simplekticheskih gruppah. Moscow: Mir Publ., 1979, 167 p.

Cite this article as: V.M.Sinitsin. On genetic codes of certain groups with 3-transpositions, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 184–188.