Further observations are made on the author's earlier paper (Ref. Zh. Mat., 1977, 5A284) dealing with the lattice H of all subgroups of the full linear group GL(n, K) over a field K that contain the group K of diagonal matrices. It is noted, for example, that for an infinite field K all subgroups inD(n, K) are algebraic; a subgroup in H is connected if and only if it is a net subgroup; the lattice of all connected subgroups in H is isomorphic to the lattice of all marked topologies onn points; any subgroup H in H is a semidirect product H=A·Ho of a maximal connected normal subgroup Ho of H and a finite group A of, permutation matrices.
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Z. I. Borevich, “A description of the subgroups of the full linear group that contain the group of diagonal matrices,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,64, 12–29 (1976).
Z. I. Borevich and N. A. Vavilov, “Subgroups of the full linear group over a scalarsemilocal ring that contain the group of diagonal matrices,” Tr. Mat. Inst. Akad. Nauk SSSR,148 (1977).
V. A. Koibaev, “Examples of nonmonomial linear groups without transvections,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,71, 153–154 (1977).
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 42–46, 1977.
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Borevich, Z.I. Some subgroups of the full linear group. J Math Sci 20, 2528–2532 (1982). https://doi.org/10.1007/BF01681469
- Normal Subgroup
- Finite Group
- Early Paper
- Linear Group
- Diagonal Matrice