Lattice Of Normal Subgroups Research Articles

Commutativity Criterions Using Normal Subgroup Lattices

We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B × G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class of all groups by the lattice of all normal subgroups of some groups, e.g. if A is an Abelian group and G is a group such that \mathbb Z × A and \mathbb Z × G have isomorphic normal subgroup lattices then A and G are isomorphic groups.

Pro- p groups with few normal subgroups

Motivated by the study of pro- p groups with finite coclass, we consider the class of pro- p groups with few normal subgroups. This is not a well defined class and we offer several different definitions and study the connections between them. Furthermore, we propose a definition of periodicity for pro- p groups, thus, providing a general framework for some periodic patterns that have already been observed in the existing literature. We then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups, but in addition have periodicity in the lattice of normal subgroups.

Lattice of normal subgroups of a group of local isometries of the boundary of a spherically homogeneous tree

We describe the structure of the lattice of normal subgroups of the group of local isometries of the boundary of a spherically homogeneous tree LIsom ∂T. It is proved that every normal subgroup of this group contains its commutant. We characterize the quotient group of the group LIsom ∂T by its commutant.

Normal Subgroups of B u Aut(Ω)

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B u Aut(Ω) has \(2^{2^{\aleph_0}}\) normal subgroups, and there is a certain ideal \({\cal Z}\) of the lattice of subsets of Open image in new window (\(\mathbb{Z}\)), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to \({\cal Z}\).

Distributive congruence lattices of congruence-permutable algebras

We prove that every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ 1 bound is optimal, as we find a distributive algebraic lattice D with ℵ 2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither of any group nor of any module), thus solving negatively a problem of E.T. Schmidt from 1969. Furthermore, D may be taken as the congruence lattice of the free bounded lattice on ℵ 2 generators in any non-distributive lattice variety. Some of our results are obtained via a functorial approach of the semilattice-valued ‘distances’ used by B. Jónsson in his proof of Whitman's Embedding Theorem. In particular, the semilattice of compact elements of D is not the range of any distance satisfying the V-condition of type 3/2. On the other hand, every distributive 〈 ∨ , 0 〉 -semilattice is the range of a distance satisfying the V-condition of type 2. This can be done via a functorial construction.

The lattice of intuitionistic fuzzy congruences

First, we prove that the set of intuitionistic fuzzy congruences on a semigroup satisfying the particular condition is a modular lattice [Theorem 2.9]. Secondly, we prove that the set of all intuitionistic fuzzy congruences on a regular semigroup contained in (χH, χHc) forms a modular lattice [Proposition 3.5]. And also we show that the set of all intuitionistic fuzzy idempotent separating congruences on a regular semigroup forms a modular lattice[Theorem 3.6]. Moreover, we prove that the lattice of intuitionistic fuzzy congruences on a regular semigroup is a disjoint union of some modular sublattices of the lattice[Corollary 3.15]. Finally, we show that the lattice of intuitionistic fuzzy congruences on a group and the lattice of intuitionistic fuzzy normal subgroups satisfying the particular condition are lattice isomorphic[Theorem 4.6]. Corresponding author 212 Kul Hur, Su Youn Jang and Hee Won Kang Mathematics Subject Classification: 03F55, 06B10, 06C05

REDUCING FUZZY ALGEBRA TO CLASSICAL ALGEBRA

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras. The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets. The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀. The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

On tensor factorisation for representations of finite groups

We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G.

Tom Head's join structure of fuzzy subgroups

Tom Head formulated a “Metatheorem” in 1995 [Fuzzy Sets and Systems 73 (1995) 349] with an idea of directly proving the fuzzy analogs of results of classical Mathematics. In the process he defined a vital concept of tip extended pair of fuzzy subgroups. In this paper we use this concept to define the join of fuzzy subgroups. Using this definition of join, we have provided a much simpler and direct proof of modularity of the lattice of fuzzy normal subgroups. Moreover, we have also demonstrated other applications regarding quasinormality and fuzzy submonoid generated by two fuzzy submonoids using the redefined sup-min product due to Kim [Inform. Sci. 91 (1996) 77].

The Top of the Lattice of Normal Subgroups of the Grigorchuk Group

A complete description of the lattice of all normal subgroups not contained in the stabilizer of the fourth level of the tree and, consequently, of index ≤212 in the Grigorchuk group G is given. This leads to the following sharp version of the congruence property: a normal subgroup not contained in the stabilizer at level n+1 contains the stabilizer at level n+3 (in fact such a normal subgroup contains the subgroup Nn+1), but, in general, it does not contain the stabilizer at level n+2. The determination of all normal subgroups at each level n≥4 is then reduced to the analysis of certain G-modules which depend only on n and the previous description, as for the analogous problem for the automorphism group of the regular rooted tree.

The automorphism group of the universa distributive lattice

We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index less than \( 2^{\aleph_0} \) lie between the pointwise and the setwise stabilizer of a finite set.

The join of fuzzy algebraic substructures of a group and their lattices

This is a continuation of work in previous papers [N. Ajmal and K.V. Thomas, Fuzzy Sets and Systems 58 (1993) 217; Inform. Sci. 76 (1994) 1]. Here, we provide a common technique of constructing the join of fuzzy substructures. Consequently, it leads to the formation of various types of lattices and sublattices of fuzzy substructures of a group including those of normal and quasinormal fuzzy subgroups. A lattice theoretic relationship of these sublattices is established. Moreover, we present simple and direct proofs for the fact that the lattice of fuzzy normal subgroups is modular.

Embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras

Given a universal algebra A, we define an algebra A′ where A and A′ are of the same type and generate the same variety of algebra. Next we define a map e: LF( A) → LC( A′) where LF( A) is the lattice of fuzzy subalgebras of A and LC( A′) is the lattice of crisp subalgebras of A′. For many algebras A, the map e is shown to be a lattice embedding. As a corollary of this fact we derive the results that the lattice of fuzzy normal subgroups of a group, and the lattice of fuzzy ideals of a ring are both modular lattices.

P-groups in the Betti-Mathieu group

We construct a Sylow p-subgroup of the Betti-Mathieu group by characterizing a certain type of polynomial through the use of the trace function and a normal basis over a finite field. It follows from this that a complete lattice of normal subgroups of the Sylow p-subgroup under inclusion will be determined together with an explicit polynomial representation of the normal subgroups.

THIN GROUPS OF PRIME-POWER ORDER AND THIN LIE ALGEBRAS

IN this paper we extend the theory of thin groups of prime-power order in a direction suggested by the theory of groups of prime-power order with fixed co-class, [9,14]. Moreover, we have found it helpful to use associated Lie algebras in this study. Let p be a prime. Recall that a group of p-power order is thin [2] if all the anti-chains in its lattice of normal subgroups are short. Here an anti-chain in a lattice is a subset of pairwise incomparable elements; and by saying that anti-chains are short we mean that they have length at most p +1. In fact, the definition extends to pro-/? -groups, and their lattices of (closed) normal subgroups. Thus p -groups of maximal class, or co-class 1, are examples of thin groups; for the theory of groups of co-class 1 see [10]. Examples of thin infinite pro-p-groups are the binary p-adic group, [7, 111.17], and the Nottingham group [9,15]. Brandl, Caranti and Scoppola [3] have shown that metabelian thin p-groups have order at most p?. It is clear that the lattice of normal subgroups of a non-(pro-)cyclic (pro-)p-group has anti-chains of length p +1 . To avoid trivialities, it is convenient to exclude (pro-)cyclic groups from our consideration of thin groups here. The only thin abelian p-group is thus elementary of order p; its lattice of normal subgroups we refer to as a diamond. It follows that the commutator factor group of a thin group is elementary of order p, so that thin groups are 2-generator groups, and every factor of the lower central series has exponent p and order at most p. It also follows (see [2,3] and the next section) that every non-trivial normal (closed) subgroup in a thin group lies between two consecutive terms of the lower central series so that in a thin pro-p-group all normal subgroups are of finite index, that is, open. Thus the lattice of normal subgroups of a thin group looks like a sequence of diamonds linked by chains. The groups of

A metatheorem for deriving fuzzy theorems from crisp versions

A metatheorem is developed that allows theorems containing references to crisp substructures to be reformulated to refer to fuzzy substructures. Theorems concerning fuzzy substructures in semigroups, rings, and lattices are derived in concise fashion from analogous theorems that concern crisp substructures. The modularity of the lattice of fuzzy normal subgroups of a group is concisely derived from the modularity of its lattice of crisp normal subgroups. The fundamental tool used is a function that represents the algebra of fuzzy subsets of a set as a subdirect product of copies of the algebra of crisp subsets.

The lattice of fuzzy normal subgroups is modular

We have introduced and discussed in a recent paper [2] various sublattices such as L f , L t , and L f n t , of the lattice L of fuzzy subgroups of a given group. In particular, we proved that the sublattice L f n , of fuzzy normal subgroups with finite range and tip “ t” is a modular sublattice of L. Here, we remove all the restrictions on the choice of fuzzy normal subgroups, and prove that the class of all fuzzy normal subgroups Ln forms the modular sublattice of L. As a tool, we prove an important result that the fuzzy subgroups generated by a pair of fuzzy normal subgroups of a group is fuzzy normal.

A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups

In this paper, we introduce the concepts of t- fuzzy congruences and t- fuzzy equivalences. Using these ideas, we investigate completely, on one hand, the lattice structures of the set of fuzzy equivalence relations on a group and the set of fuzzy congruences and, on the other hand, the lattice structures of the set of fuzzy subgroups and fuzzy normal subgroups. Our study reveals some finer and interesting facts about these lattices. It is proved, among other results, that the set Ct of all t- fuzzy congruences of a group G forms lattice, and also the set L n t of all those fuzzy normal subgroups, which assume the same value t at e the identity of G, forms a lattice. As an important result, we prove that the lattices C t and L n t are isomorphic. It is also shown that the lattices C t and L n t are modular. Moreover, we construct various important sublattices of the lattice C t and exhibit their relationship by lattice diagrams. In the process, we improve and unify many results of earlier authors on fuzzy congruences.

Varieties of groups and normal-subgroup lattices?a survey

In this paper we bring together a number of results on a problem of Jonsson about characterising a variety of groups by means of the normal-subgroup lattices of its member groups.

The lattices of fuzzy subgroups and fuzzy normal subgroups

Here we discuss various types of sublattices of the lattice of fuzzy subgroups of a given group. We prove that a special class of fuzzy normal subgroups constitutes a modular sublattice of the lattice of fuzzy subgroups.