Maximal subgroups of the orthogonal group

Let V be an n-dimensional regular quadratic space over a field K of characteristic not 2. Assume n 4. Let W be a regular hyperplane and v a nonzero vector orthogonal to W. Suppose every regular hyperplane in W is universal. If is an isometry of V not leaving W invariant, then , together with the isometries of W, generate the orthogonal group of V, with one exception.The work of the author was partially supported by NSERC Grant A-7862.     

Tensor invariants for certain subgroups of the orthogonal group

Let V be an n-dimensional vector space, and let O _n be the orthogonal group. Motivated by a question of B. Szegedy (J. Am. Math. Soc. 20(4), 2007), about the rank of edge connection matrices of partition functions of vertex models, we give a combinatorial parameterization of tensors in V ^?k invariant under certain subgroups of the orthogonal group. This allows us to give an answer to this question for vertex models with values in an algebraically closed field of characteristic zero.     

A characterization of fuzzy subgroups

An ordinary subgroup of a group G is (1) a subset of G, (2) closed under the group operation. In a fuzzy subgroup it is precisely these two notions that lose their deterministic character. A fuzzy subgroup μ of a group (G,·) associates with each group element a number, the larger the number the more certainly that element belongs to the fuzzy subgroup. The closure property is captured by the inequality μ(x · y)?T(μ(x), μ(y)). In A. Rosenfeld's original definition, T was the function ‘minimum’. However, any t-norm T provides a meaningful generalization of the closure property. Two classes of fuzzy subgroups are investigated. The fuzzy subgroups in one class are subgroup generated, those in the other are function generated. Each fuzzy subgroup in these classes satisfies the above inequality with T given by T(a, b) = max(a + b ?1, 0). While the two classes look different, each fuzzy subgroup in either is isomorphic to one in the other. It is shown that a fuzzy subgroup satisfies the above inequality with $\text{T =}\text{‘minimum’}$ if and only if it is subgroup generated of a very special type. Finally, these notions are applied to some abstract pattern recognition problems.     

Jordan subgroups and orthogonal decompositions

Translated from Algebra i Logika, Vol. 28, No. 4, pp. 382–392, July–August, 1989.     

COMPLETENESS OF THE PARABOLIC SUBGROUPS OF PROJECTIVE ORTHOGONAL GROUPS

All parabolic subgroups and Borel subgroups of PΩ(2m 1, F) over a linear-able field F of characteristic 0 are shown to be complete groups, provided m > 3.     

Congruence subgroups of the symplectic group

Congruence subgroups closely related to parabolic subgroups are described in the symplectic group over a Dedekind ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 80–91, 1976.     

Number theoretic characterization of the subgroups and factor groups of a finite cyclic group

This paper examines the structure of the subgroups and factor groups of finite cyclic groups by using number theoretic techniques. Such number theoretic concepts as the g.c.d., the 1.c.m., and the Euler phi function are used to characterize various subgroups and factor groups. Results relative to the orders of subgroups are also developed. It is also shown that certain group properties yield number theoretic results.     

Characterized subgroups of the circle group

Ricerche di Matematica - A subgroup H of the circle group $$\mathbb T$$ is said to be characterized by a sequence $$\mathbf {u}= (u_n)_{n\in \mathbb N}$$ of integers if $$H=\{x\in \mathbb T:...     

Equality of orthogonal transvection group and elementary orthogonal transvection group

H. Bass defined orthogonal transvection group of an orthogonal module and elementary orthogonal transvection group of an orthogonal module with a hyperbolic direct summand. We also have the notion of relative orthogonal transvection group and relative elementary orthogonal transvection group with respect to an ideal of the ring. According to the definition of Bass relative elementary orthogonal transvection group is a subgroup of the relative orthogonal transvection group of an orthogonal module with hyperbolic direct summand. Here we show that these two groups are the same in the case when the orthogonal module splits locally.     

A lattice theoretic characterization of prefrattini subgroups

Consider an interval [H,G] in the lattice of subgroups of a finite soluble groupG. We define a certain set of subgroups in the lattice [H,G], and prove that they are conjugate inG. ForH=1 one gets the prefrattini subgroups ofG.     

A study of normal fuzzy subgroups and characteristic fuzzy subgroups of a fuzzy group

In this paper, we investigate the properties of normal fuzzy subgroups of a fuzzy group. We also introduce the notion of a characteristic fuzzy subgroup of a fuzzy group, of which we provide level subset and strong level subset characterizations. Then we prove that a characteristic fuzzy subgroup of a fuzzy group is a normal fuzzy subgroup. Besides we prove that the commutator subgroup generated by the commutator of two normal fuzzy subgroups of a fuzzy group is contained in their intersection. Finally, we construct many new lattices and sublattices of fuzzy subgroups and normal fuzzy subgroups of a given fuzzy group. We also construct lattices of characteristic fuzzy subgroups possessing sup-property.     

Normal subgroups in the Cremona group

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $ \mathbb{P}_{\mathbf{k}}^2 $ is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.