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We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.As a rule, the quality of descriptions of the subnormal subgroups of linear groups is determined by some arithmetic function f (d) of the subnormal depth d of a subgroup H in a group G: the smaller the value of f (d), the better the description. For the general linear groups and symplectic groups in all dimensions the best available functions are f (d) = 1 4 (5 d − 1) and f (d) = 1 5 (6 d − 1); see [1][2][3][4][5][6]. In this article we improve these functions for rather wide classes of ground rings. In Sections 1 and 2 we obtain the function f (d) = 1 3 (4 d − 1) for the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible. In Section 3 we obtain the function f (d) = 1 4 (5 d − 1) for the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.The row of a matrix over some ring R whose elements generate the entire ring R as an ideal is called unimodular. Take some integers m ≥ 1 and n ≥ 2. Following [7] call a commutative ring R (m, n)-full if given an (m × n)-matrix A over R with unimodular rows there exist an element ε ∈ R (which depends on A) and invertible elements ε 1 , ε 2 , . . . , ε m ∈ R such thatwhere t denotes the taking of transpose. It is clear that an (m, n)-full ring is also (s, t)-full for s ≤ m and t ≤ n. Therefore, each (m, n)-full ring is a (1, 2)-full ring; i.e., a ring of stable rank 1. It is known (see [7]) that if each residue field of a semilocal or dimension 0 ring, which in particular includes von Neumann regular rings, contains more than m(n − 1) elements then this ring is (m, n)-full.A unital commutative ring with the only maximal ideal is called a local ring. Recall that if there exists a seriesof subgroups of some group G, where H i−1 H i means that H i−1 is a normal subgroup of H i , then H is called a subnormal subgroup of G. In this case we write H d G. The smallest of these d is called the subnormal depth of H in G. Take some ring R of stable rank 1 (in particular, every full or local ring will do) in which 2 is invertible, and some ideal I of R. The canonical ring homomorphism R → R/I induces the surjective [7] group homomorphism π I : GL 2 (R) → GL 2 (R/I). Here by GL 2 (R/R) we mean the trivial group. Put Z I = (center GL 2 (R/I))π −1 I , K I = {σ ∈ SL 2 (R) | σπ I = 1},where t ij are transvections. Note that Z R = GL 2 (R), Z 0 = center GL 2 (R), K R = SL 2 (R), and K 0 = {1}. Call the ideal J(h) = id(a − d, b, c) the weight of h = a b c d , and define the weight of a subgroup H as J(H) = h∈H J(h). Denote by i n an arbitrary element of the ideal I n . Denote by R * the group of invertible elements of a ring R.Nukus.
We describe the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible, as well as the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.As a rule, the quality of descriptions of the subnormal subgroups of linear groups is determined by some arithmetic function f (d) of the subnormal depth d of a subgroup H in a group G: the smaller the value of f (d), the better the description. For the general linear groups and symplectic groups in all dimensions the best available functions are f (d) = 1 4 (5 d − 1) and f (d) = 1 5 (6 d − 1); see [1][2][3][4][5][6]. In this article we improve these functions for rather wide classes of ground rings. In Sections 1 and 2 we obtain the function f (d) = 1 3 (4 d − 1) for the subnormal subgroups of 2-dimensional linear groups over local and full rings in which 2 is invertible. In Section 3 we obtain the function f (d) = 1 4 (5 d − 1) for the subnormal subgroups of symplectic groups over local rings in which 2 is invertible.The row of a matrix over some ring R whose elements generate the entire ring R as an ideal is called unimodular. Take some integers m ≥ 1 and n ≥ 2. Following [7] call a commutative ring R (m, n)-full if given an (m × n)-matrix A over R with unimodular rows there exist an element ε ∈ R (which depends on A) and invertible elements ε 1 , ε 2 , . . . , ε m ∈ R such thatwhere t denotes the taking of transpose. It is clear that an (m, n)-full ring is also (s, t)-full for s ≤ m and t ≤ n. Therefore, each (m, n)-full ring is a (1, 2)-full ring; i.e., a ring of stable rank 1. It is known (see [7]) that if each residue field of a semilocal or dimension 0 ring, which in particular includes von Neumann regular rings, contains more than m(n − 1) elements then this ring is (m, n)-full.A unital commutative ring with the only maximal ideal is called a local ring. Recall that if there exists a seriesof subgroups of some group G, where H i−1 H i means that H i−1 is a normal subgroup of H i , then H is called a subnormal subgroup of G. In this case we write H d G. The smallest of these d is called the subnormal depth of H in G. Take some ring R of stable rank 1 (in particular, every full or local ring will do) in which 2 is invertible, and some ideal I of R. The canonical ring homomorphism R → R/I induces the surjective [7] group homomorphism π I : GL 2 (R) → GL 2 (R/I). Here by GL 2 (R/R) we mean the trivial group. Put Z I = (center GL 2 (R/I))π −1 I , K I = {σ ∈ SL 2 (R) | σπ I = 1},where t ij are transvections. Note that Z R = GL 2 (R), Z 0 = center GL 2 (R), K R = SL 2 (R), and K 0 = {1}. Call the ideal J(h) = id(a − d, b, c) the weight of h = a b c d , and define the weight of a subgroup H as J(H) = h∈H J(h). Denote by i n an arbitrary element of the ideal I n . Denote by R * the group of invertible elements of a ring R.Nukus.
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