Congruence Groups Research Articles

On the Isomorphisms between Certain Congruence Groups

In this paper we study the possible isomorphisms between the congruence subgroups of the classical groups over integral domains by applying the involution-free techniques previously used by O’Meara and the author. We prove, in dimensions at least 6 and characteristic not 2, that a linear congruence group is never isomorphic to a symplectic congruence group nor to an isotropic unitary congruence group whose associated hermitian space has Witt index at least 3.

On a sublattice of the lattice of congruences on a simple regular ω-semigroup

The set E of idemponents of a semigroup S can be partially ordered by defining e ≦f and only if ef = fe = e (e,f ∈ E). If E = {ei: i = i = 0,1…} and under this ordering e0 > e1 > e2… then we call S an ω-semigroup. Munn [10] has given a complete classification of simple regular ω-semigroups in terms of groups and group homomorphisms. Let ∧0(S) denote the set of congruences on a simple regular w-semigroup S consisting of those congruences which either are idempotent-separating or are group congruences on S. It is evident that ∧0(S) is a sublattice of the lattice of all congruences on S.

The automorphisms of the symplectic congruence groups

The automorphisms of the symplectic congruence groups

On the isomorphisms between certain congruence groups

In this paper we study the possible isomorphisms between the congruence subgroups of the classical groups over integral domains by applying the involution-free techniques previously used by O’Meara and the author. We prove, in dimensions at least 6 and characteristic not 2, that a linear congruence group is never isomorphic to a symplectic congruence group nor to an isotropic unitary congruence group whose associated hermitian space has Witt index at least 3.

Congruences et Automorphismes des Automates Finis

We study a class of congruences of strongly connected finite automata, called the group congruences, which may be defined in this way: every element fixing any class of the congruence induces a permutation on this class. These congruences form an ideal of the lattice of all congruences of the automaton $$\mathfrak{A}$$ and we study the group associated with the maximal group congruence (maximal induced group) with respect to the Suschkevitch group of the transition monoid of $$\mathfrak{A}$$ . The transitivity equivalence of the subgroups of the automorphism group of $$\mathfrak{A}$$ are found to be the group congruences associated with regular groups, which form also in ideal of the lattice of congruences of $$\mathfrak{A}$$ . We then characterize the automorphism group of $$\mathfrak{A}$$ with respect to the maximal induced group. As an application, we show that, given a group G and an automaton $$\mathfrak{A}$$ , there exists an automaton whose automorphism group is isomorphic to G and such that the quotient by the automorphism congruence is $$\mathfrak{A}$$ .

Isomorphism theory of congruence groups

This note is an announcement of four theorems I proved in [5][9] on the isomorphisms of the linear, symplectic, and unitary congruence groups. A sketch of the proofs is given. NOTATION. Let V be an w-dimensional vector space over the field F, n*z2. For crGGLw(F), let cr denote the contragredient of cr (inverse of the transpose). Let ~ denote the natural map of GLn( V) onto PGLn( V) ; for any subset 5 of GLn(V), S is the image of S in PGLn(V) under the map. A transvection r is a linear transformation of determinant one which fixes all vectors of some hyperplane, called the proper hyperplane of r. If r 5 1 then (r — 1) V is a line called the proper line of r. An element f of PGLn(V) is called a (projective) transvection if r is a transvection. The proper line and proper hyperplane of f are defined as the proper line and hyperplane of r.

On the minimal group congruence on the tensor product of Archimedean commutative semigroups

On the minimal group congruence on the tensor product of Archimedean commutative semigroups

Congruences on regular semigroups

The kernel of a congruence on a regular semigroup S may be characterized as a set of subsets of S which satisfy the Teissier-Vagner-Preston conditions. A simple construction of the unique congruence associated with such a set is obtained. A more useful characterization of the kernel of a congruence on an orthodox semigroup (a regular semigroup whose idempotents form a subsemigroup) is provided, and the minimal group congruence on an orthodox semigroup is determined.

Effects of counselor-client cognitive congruence on counseling outcome in brief counseling.

This study examined the effects of three levels of counselor-cli ent congruence (low, medium, and high) upon counseling outcome. Four staff and four graduate assistant counselors were matched with clients prior to counseling on the basis of the similarity in meaning which they assigned to 12 concepts on the Evaluation dimension of a semantic differential. An analysis of variance design revealed a significant difference (p < .05) in change between clients of low, medium, and high congruency, with low-congruence clients changing significantly and more consistently than the other congruence groups. Change in congruence was significantly different in dyads of low, medium, and high congruence (p < .05). Staff counselors had more client-congruence change associated with their counseling.

Lattice subgroups of free congruence groups

Let Г(1) denote the homogeneous modular group of 2 × 2 matrices with integral entries and determinant 1. Let (1) be the inhomogeneous modular group of 2 × 2 integral matrices of determinant 1 in which a matrix is identified with its negative. (N), the principal congruence subgroup of level N, is the subgroup of (1) consisting of all T ∈ (1) for which T ≡ ± I (mod N), where N is a positive integer and I is the identity matrix. A subgroup of (1) is said to be a congruence group of level N if contains (N) and N is the least such integer. Similarly, we denote by Г(N) the principal congruence subgroup of level N of Г(1), consisting of those T∈(1) for which T ≡ I (mod N), and we say that a sub group of Г(1) is a congruence group of level N if contains Г (N) and N is minimal with respect to this property. In a recent paper [9] Rankin considered lattice subgroups of a free congruence subgroup of rank n of (1). By a lattice subgroup of we mean a subgroup of which contains the commutator group . In particular, he showed that, if is a congruence group of level N and if is a lattice congruence subgroup of of level qr, where r is the largest divisor of qr prime to N, then N divides q and r divides 12. He then posed the problem of finding an upper bound for the factor q. It is the purpose of this paper to find such an upper bound for q. We also consider bounds for the factor r.

Isometric Circles of Congruence Groups

Isometric Circles of Congruence Groups

The automorphisms of the linear congruence groups over Dedekind domains

Let o be a Dedekind domain with quotient field F and let M be an o -lattice on an n-dimensional vector space V over F. Let G be any one of the linear congruence groups TL n(M;a), Sl n(M;a),GL n(M;a) determined by an integral ideal a. If n ≥ 3, then every automorphism Λ of G can be uniquely expressed in exactly one of the forms ∧=P x°Φ g or ∧=P x°Psi; h where P χ is a radial automorphism, Φ g is associated with certain semilinear isomorphisms of V onto V and Ψ h is associated with certain semilinear isomorphisms of V onto the dual space of V. For background and motivation see O. T. O'Meare, The automorphisms of the linear groups over any integral domain, J. reine angew. Math. 223 (1966), pp. 56–100. The proof uses results on the characterization of transvections by O. T. O'Meara (Group-theoretic characterization of transvections using CDC subgroups) and by H. Zassenhaus (Characterization of unipotent matrices).

Two-Dimensional Linear Groups Over Local Rings

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 .

Group congruences and quasifields

Group congruences and quasifields

Maximal normal subgroups of the modular group

Furthermore, those of level $p are not congruence groups. It is well known that r contains infinitely many normal of finite index which are not congruence groups; see for example papers [3], [4]. However, all of these groups have the common feature that they are lattice subgroups (in Rankin's terminology) of some normal congruence group, and so are not maximal. The results of this paper imply that r contains infinitely many maximal normal of finite index which are not congruence groups, a somewhat surprising fact. The question which originally motivated this paper was the following: Which of the known simple groups have a representation as a modular quotient group, or equivalently, may be generated by two elements, one of period 2, the other of period 3? Call such a group a r-group. Then the groups LF(2, p) are certainly r-groups. However, it is not known for example when the alternating group is a r-group. In his lecture notes on Fuchsian groups [2], Macbeath poses a similar question for H-groups, and makes some remarks about the linear fractional groups which in fact form the basis of this paper, and which we are happy to acknowledge here.

Lattice subgroup of free congruence groups

Lattice subgroup of free congruence groups

Irreducible representations of the binary modular congruence groups mod $p^{\lambda}$

Irreducible representations of the binary modular congruence groups mod $p^{\lambda}$

Isomorphic congruence groups and Hecke operators

Let G, H, K be groups such that G is normal in K and G ⊆ H ⊆ K. Let I(H, K) be the set of inner automorphisms of K restricted to H; thus α ∊ I(H, K) if and only if, for some κ∊ K, α(h) = k-1hk for all h ∊ H. Let φ be an isomorphism of H/G onto a subgroup Hφ/G of K/G. An isomorphism Φ of H onto H(φ) is called an extension of ø ifΦ(h)G = φ(hG) for all h∊H.

Certain fundamental congruences on a regular semigroup

In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some restriction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating congruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall consistently denote by α β and η respectively.

Normal subgroups of the modular group which are not congruence subgroups

A subgroup of r containing a principal congruence subgroup r(n) is said to be a congruence subgroup, and is of level n if n is the least such integer. In a recent article [2] the writer determined all normal subgroups of r of genus 1 (see [1 ] for the definition of the genus of a subgroup of r). An interesting question that arises is to decide which of these are also congruence subgroups. In this note we show that there are just 4 such groups. This furnishes a new family of normal subgroups of finite index in the modular group which are not congruence groups (see [3 ] for other examples). The proof makes use of some recent work of K. Wohlfahrt [4] on the definition of level for an arbitrary subgroup of finite index in r. We let x stand for the substitution