On Irreducible Carpets of Additive Subgroups of Type G2
This article discusses the subgroups of Chevalley groups, defined by carpets — the sets of additive subgroups of the main definition ring. Such subgroups are called carpet subgroups and they are generated by root elements with coefficients from the corresponding additive subgroups. By definition, a carpet is closed if the carpet subgroup, which it defines, does not contain new root elements. One of the fundamentally important issues in the study of carpet subgroups is the problem of the closure of the original carpet. It is known that this problem is reduced to irreducible carpets, that is, to carpets, all additive subgroups of which are nonzero [8;11]. This paper describes irreducible carpets of type G2 over a field K of characteristics p > 0, all additive subgroups of which are R–modules, in case when K is an algebraic extension of R. It is proved that such carpets are closed and can be parametrized by two different fields only for p = 3, and for other p they are determined by one field. In this case the corresponding carpet subgroups coincide with Chevalley groups of type G2 over intermediate subfields P, R ⊆ P ⊆ K.
Svetlana Franchuk (Kuklina), Postgraduate, Siberian Federal University, 79, Svobodniy pr., Krasnoyarsk, 660041, Russian Federation, e-mail: email@example.com
Franchuk S.K. On Irreducible Carpets of Additive Subgroups of Type G2. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 27, pp. 80-86. (in Russian) https://doi.org/10.26516/1997-7670.2019.27.80
- Dryaeva R.Y., Koibaev V.A., Nuzhin Ya.N. Full and elementary nets over the quotient ﬁeld of a principal ideal ring. Zap. Nauchn. Sem. POMI, 2017, vol. 455, pp. 42-51. (in Russian)
- Koibaev V.A. Elementary nets in linear groups. Tr. IMM UrB RAS, 2011, vol. 17, no. 4, pp. 134–141. (in Russian)
- Koibaev V.A., Kuklina S.K., Likhacheva A.O., Nuzhin Ya.N. Subgroups, of Chevalley groups over a locally ﬁnite ﬁeld, deﬁned by a family of additive subgroups. Mathematical Notes, 2017. vol. 102, pp. 857-865. (in Russian) https://doi.org/10.4213/mzm11038
- Koibaev V.A., Nuzhin Ya.N. Subgroups of the Chevalley groups and Lie rings deﬁnable by a collection of additive subgroups of the initial ring. Fundam. Prikl. Mat., 2013, vol. 18, no. 1, pp. 75-84. (in Russian)
- Koibaev V.A., Nuzhin Ya.N. k-invariant nets over an algebraic extension of a ﬁeld k. Sib Math Journal, 2017, vol. 58, no. 1, pp. 143–147. (in Russian) https://doi.org/10.17377/smzh.2017.58.114
- Kuklina (Franchuk) S.K. On irreducible carpets of additive subgroups of type G2 . International Algebraic Conference dedicated to the 110th anniversary of Professor A. G. Kurosh. Moscow, MSU Publ., 2018. pp. 247-248.
- Kuklina S.K., Likhacheva A.O., Nuzhin Ya.N. On closeness of carpets of Lie type over commutative rings. Tr. IMM UrB RAS, 2015, vol. 21, no. 3, pp. 192–196. ( in Russian)
- Levchuk V.M. On generating sets of root elements of Chevalley groups over a ﬁeld. Algebra and logica, 1983, vol. 22, no. 5, pp. 504-517. (in Russian)
- Levchuk V.M. Parabolic subgroups of certain ABA-groups. Mathematical Notes, 1982, vol. 31, no. 4, pp. 509-525. (in Russian)
- Nuzhin Ya.N. About subgroups of Chevalley groups of type Bl, Cl, F4 and G2 parametrized by two imperfectﬁelds of characteristic 2 and 3. Mathematics in the modern world, 2017, p. 90. (in Russian)
- Nuzhin Ya.N. Levi decomposition for carpet subgroups of Chevalley groups over a ﬁeld. Algebra i logica, 2016, vol. 55, no. 5, pp. 558-570. (in Russian) https://doi.org/10.17377/alglog.2016.55.503
- Nuzhin Ya.N. Factorization of carpet subgroups of the Chevalley groups over commutative rings. J. Sib. Fed. Univ. Math. Phys, 2011, vol. 4, no. 4, pp. 527-535. (in Russian)
- Steinberg R. Lectures on Chevalley Groups Moscow, Mir Publ., 1975. (in Russian)
- Carter R.W. Simple groups of Lie type. Pure Appl. Math., 1972. No. 28.