Sandwich classification theorem

Document Type: Research Paper

Author

St.Petersburg State University

Abstract

‎The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''‎. ‎For subgroups $F\le N$ of a group $G$ denote by $L(F,N)$‎ ‎the set of all subgroups of $N$‎, ‎containing $F$‎. ‎Let $D$ be a subgroup of $G$‎. ‎In this note we study the lattice $LL=L(D,G)$ and the lattice $LL'$‎ ‎of subgroups of $G$‎, ‎normalized by $D$‎. ‎We say that $LL$ satisfies sandwich classification‎ ‎theorem if $LL$ splits into a disjoint union of sandwiches $L(F,N_G(F))$‎ ‎over all subgroups $F$ such that the normal closure of $D$ in $F$ coincides with $F$‎. ‎Here $N_G(F)$ denotes the normalizer of $F$ in $G$‎. ‎A similar notion of sandwich classification‎ ‎is introduced for the lattice $LL'$‎. ‎If $D$ is perfect‎, ‎i.\,e‎. ‎coincides with its commutator‎ ‎subgroup‎, ‎then it turns out that sandwich classification theorem for $LL$ and $LL'$ are equivalent‎. ‎We also show how to find basic subroup $F$ of sandwiches for $LL'$ and review‎ ‎sandwich classification theorems in algebraic groups over rings‎.

Keywords

Main Subjects


E. Abe (1989). Normal subgroups of Chevalley groups over commutative rings. Amer. Math. Soc.. 83, 1-17
A. S. Ananevskii, N. A. Vavilov and S. S. Sinchuk (2012). Overgroups of ( l ; R ) E( m; R ) I. Levels and normalizers. St. Petersburg Math. J.. 23, 819-849
A. S. Ananievskii, N. A. Vavilov and S. S. Sinchuk (2009). Overgroups of ( l ; R ) ( m; R ), J. Math. Sci. (N. Y.). 161, 461-473
M. S. Ba and Z. I. Borevichf (1988). On the distribution of intermediate subgroups. (Russian) Rings and linear groups, Kubansky State University, Krasnodar. , 14-41
S. V. Bakulin and N. A. Vavilov (2011). On subgroups normalized by EO(2 l ; R ). Vestnik St. Petersburg Univ. Math.. 44, 252-259
H. Bass (1964). K -theory and stable algebra. Inst. Hautes tudes Sci. Publ. Math.. 22, 5-60
Z. I. Borevich (1981). A description of the subgroups of the complete linear group that contain the group of diagonal matrices. J. Soviet Math.. 17, 1718-1730
Z. I. Borevich (1982). On arrangement of subgroups. J. Soviet Math.. 19, 977-981
Z. I. Borevich and O. N. Macedonska ja (1984). On the lattice of subgroups. J. Soviet Math.. 24, 395-399
Z. I. Borevich and N. A. Vavilov (1980). Subgroups of the general linear group over a semilocal ring. containing the group of diagonal matrices , Pro c. Steklov. Inst. Math.. 148, 41-54
Z. I. Borevich and N. A. Vavilov (1982). The distribution of subgroups containing a group of blo ck diagonal matrices in the general linear group over a ring. Sov. Math.. 26, 13-18
Z. I. Borevich and N. A. Vavilov (1985). The distribution of subgroups in the general linear group over a commutative ringc. Proc. Steklov. Inst. Math.. 165, 27-46
I. Z. Golub chik (1973). On the general linear group over an asso ciative ring. Uspekhi Math. Nauk. 27, 179-180
I. Z. Golub chik (1984). On subgroups of the general linear group GL_n ( R ) over an associativering R. Russian Math. Surveys. 39, 157-158
Ya. N. Nuzhin (1983). Groups contained b etween groups of Lie typ e over different elds. Algebra Logic. 22, 378-389
Ya. N. Nuzhin (2013). Intermediate subgroups in the Chevalley groups of type B_l , C_l , F_4 and G_2 over the nonp erfect fields of characteristic 2 and 3. Sib. Math. J.. 54, 119-123
Ya. N. Nuzhin and A. V. Yakushevich (2000). Intermediate subgroups of Chevalley groups over a field of fractions of a ring of principal ideals. Algebra and Logic. 39, 199-206
N. S. Romanovskii (1967). Maximal subrings of the field Q and maximal subgroups of the group SL( n; Q ). Algebra i Logika. 6, 75-82
A. Shchegolev (2014). Overgroups of symplectic subsystem subgroups in symplectic groups over commutative rings. Preprint.
R. A. Shmidt (1982). Subgroups of the general linear group over the field of quotients of a Dedekind ring. J. Soviet Math.. 19, 1052-1059
A. V. Stepanov (1993). On the distribution of subgroups normalized by a fixed subgroup. J. Soviet Math.. 64, 769-776
A. V. Stepanov (2010). Free pro duct subgroups b etween Chevalley groups G(; F ) and G(; F [ t]). J. Algebra. 324, 1549-1557
A. V. Stepanov (2012). Subring subgroups in Cheval ley groups with doubly laced root systems. J. Algebra. 362, 12-29
L. N. Vaserstein (1981). On the normal subgroups of GL n over a ring. Lecture Notes in Math., Springer, Berlin-New York. 854, 456-465
L. N. Vaserstein (1986). On normal subgroups of Chevalley groups over commutative rings. Tohoku Math. J. (2). 38, 219-230
N. Vavilov and V. Petrov (2003). Overgroups of EO(2l ; R ). J. Math. Sci. (N. Y.). 116 (1), 2917-2925
N. Vavilov and V. Petrov (2004). Overgroups of Ep(2 l ; R ). St. Petersburg Math. J.. 15, 515-543
N. Vavilov and V. Petrov (2008). Overgroups of EO( n; R ). St. Petersburg Math. J.. 19 (2), 167-195
N. A. Vavilov (1981). Description of subgroups of the full linear group over a semilo cal ring that contain the group of diagonal matrices. J. Soviet Math.. 17, 1960-1963
N. A. Vavilov (1995). Intermediate subgroups in Cheval ley groups. London Math, So c. Lecture Note Ser., Cambridge Univ. Press, Cambridge. 207, 233-280
N. A. Vavilov (2004). Subgroups of split orthogonal groups over a commutative ring. J. Math. Sci. (N. Y.). 120, 1501-1512
N. A. Vavilov (2008). Subgroups of symplectic groups that contain a subsystem subgroup. J. Math. Sci. (N. Y.). 151, 2937-2948
N. A. Vavilov and A. V. Shchegolev (2013). Overgroups of subsystem subgroups in exceptional groups: levels. J. Math. Sci. (N. Y.). 192, 164-195
J. S. Wilson (1972). The normal and subnormal structure of general linear groups. Proc. Cambridge Philos. Soc.. 71, 163-177
Y. Hong (2004). Overgroups of symplectic group in linear group over commutative rings. J. Algebra. 282, 23-32
Y. Hong (2006). Overgroups of classical groups in linear group over Banach algebras. J. Algebra. 304, 1004-1013
Y. Hong (2006). Overgroups of classical groups over commutative ring in linear group. Sci. China Ser. A. 49, 626-638