OVERGROUPS OF SYMPLECTIC GROUP IN LINEAR GROUP OVER LOCAL RINGS

2001 ◽  
Vol 29 (6) ◽  
pp. 2313-2318 ◽  
Author(s):  
Hong You ◽  
Baodong Zheng
Keyword(s):  
Linear Group ◽  
Symplectic Group ◽  
Local Rings

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

ON INCIDENCE MODULO IDEAL RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 553-586 ◽  
Author(s):  
M. A. DOKUCHAEV ◽  
V. V. KIRICHENKO ◽  
B. V. NOVIKOV ◽  
A. P. PETRAVCHUK
Keyword(s):  
Linear Group ◽  
Partially Ordered Set ◽  
General Linear Group ◽  
Associative Ring ◽  
Ordered Set ◽  
General Linear ◽  
Local Rings ◽  
Partially Ordered ◽  
Finite Partially Ordered Set

For a given associative ring B, a two-sided ideal J ⊂ B and a finite partially ordered set P, we study the ring A = I(P, B, J) of incidence modulo J matrices determined by P. The properties of A involving its radical and quiver are investigated, and the interaction of A with serial rings is explored. The category of A-modules is studied if P is linearly ordered. Applications to the general linear group over some local rings are given.

IRREDUCIBLE COMPLEX CHARACTERS OF THE FULL AFFINE GROUP

1995 ◽  
Vol 05 (01) ◽  
pp. 1-5
Author(s):  
M.R. DARAFSHEH ◽  
M. RAJABI TARKHORANI ◽  
A. DANESHKHAH
Keyword(s):  
Linear Group ◽  
Symplectic Group ◽  
General Linear Group ◽  
Affine Group ◽  
Character Table ◽  
General Linear ◽  
Complex Character ◽  
Irreducible Characters ◽  
Entire Complex ◽  
Complex Characters

In this paper we describe how the entire complex character table of the affine subgroup of the general linear group Hn can be constructed inductively. In particular a number of certain irreducible characters of Hn which were studied by Gow [5] are obtained. We also describe some irreducible characters of the affine symplectic group. In all cases the powerful and interesting method of B. Fischer is employed.

𝜔+-Type Index of GL+(2)-Paths

2019 ◽  
Vol 19 (4) ◽  
pp. 771-778
Author(s):  
Sai Liu ◽  
Wei Wang
Keyword(s):  
Periodic Solution ◽  
Linear Stability ◽  
Linear Group ◽  
Symplectic Group ◽  
General Linear Group ◽  
Index Theory ◽  
Homotopy Classification ◽  
Differential Systems ◽  
Formula Type ◽  
Solution Problem

AbstractIn this paper, we establish an {\omega^{+}}-type index theory for paths in the general linear group {\mathrm{GL}^{+}(2)}. This is done by the complete homotopy classification for such paths. We also compare this index theory with the ω index theory for paths in the symplectic group {\mathrm{Sp}(2)} and obtain a generalization of Bott formula for iterated paths in {\mathrm{GL}^{+}(2)}. As applications, the minimal periodic solution problem and the linear stability of general differential systems are studied.

Elementary symplectic orbits and improved K1-stability

2010 ◽  
Vol 7 (2) ◽  
pp. 389-403 ◽  
Author(s):  
Pratyusha Chattopadhyay ◽  
Ravi A. Rao
Keyword(s):  
Commutative Ring ◽  
Linear Group ◽  
Symplectic Group ◽  
General Linear Group ◽  
General Linear ◽  
Affine Algebras

AbstractIt is shown that the set of orbits of the action of the elementary symplectic group on all unimodular rows over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary general linear group. This result is used to improve injective stability for K1 of the symplectic group over non-singular affine algebras.

On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

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