Commutativity Criterions Using Normal Subgroup Lattices

We study D-hyperoctahedral groups, diagonal inductive limits of hyperoctahedral groups. Also, we describe the Z 2-modules of periodic sequences over diagonal limits of symmetric groups under their action on the elements of a Z 2-module by permutation of coordinates. The imprimitivity systems for D-hyperoctahedral groups are characterized, and a full description of the lattices of their normal subgroups is given.

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

Theorems in group theory whose proofs are effected by manipulating subgroups and not elements-such theorems generally refer to direct decompositions-are as a rule easily recognized as special cases of theorems in the theory of lattices. There are other theorems in the theory of groups where such a generalization is not quite obvious. Thus the problem arises to characterize those parts of the theory of groups in which the results may be expected to be special cases of theorems in the theory of lattices. Such a part of the theory of groups contains but one essentially group-theoretical statement, namely the assertion that all the facts in this theory are special cases of results in the theory of lattices. If the group G is isomorphic to every group with an isomorphic lattice of subgroups, then it may be said that the structure theory of this group G forms part of the theory of lattices. If in addition every of the lattice of the subgroups of G is induced by an of G, then the relative structure of the subgroups of G presents a problem which belongs to the theory of lattices. Thus it will be our problem to discuss the relations between the isomorphisms of groups on the one hand and the isomorphisms of the lattice of its subgroups on the other hand. The term isomorphism of the lattice of subgroups will be used in a more or less restricted sense. In its broadest meaning this term refers only to lattice properties in the accepted sense of the word, whereas the isomorphisms of the lattice of the subgroups in the more restricted sense of the term shall preserve properties like normality and numbers like the index. The modern development of the theory of lattices has been preceded by a discussion of the above problem. It has been proved in 1928 by A. Rottlaender that there exists -finite non-isomorphic groups whose lattices of subgroups are isomorphic in a rather strong sense.' Since these groups are -both not abelian,

In 1908 Professor G. A. Miller showed that if an abelian group H which involves operators whose orders exceed 2 is extended by means of an operator of order 2 which transforms each operator of H into its inverse, then the group of isomorphisms of this extended group is the holomorph of H.t The present paper discusses an elaboration of the idea embodied in Professor Miller's thoerem, the successive developments taking an abelian group H each of more general type so that in toto it is proved that if G is formed by extending H which has operators of order > 2 by a certain operator from its group of isomorphisms which transforms every one of its operators into the same power of itself and which is commutative with no operator of odd order in it, then the group of isomorphisms of G is the holomorph of H, and is a complete group if H is of odd order. If H contains no operators of even order, the certain operator is any operator from the group of isomorphisms of H effecting the stated automorphism; if H contains no operator of odd order, the certain operator must transform every operator of H into its inverse; if H contains operators of both even and odd orders (a) the order of the automorphism of the operators of odd order effected by the extending operator is to be divisible by 2n where H contains an operator of order 2n but none of order 2n+1, or else (b) the extending operator transforms into its inverse every one of H's operators whose order is a power of 2. Obviously it is not necessary that the extending operator be from its group of isomorphisms, but that it have certain properties possessed by this operator; thus, besides effecting the automorphism required, its first power commutative with all the operators of H must be the identity (from which follows that its first power appearing in H is the identity). The general method used in establishing each of the successive theorems is the same; two important steps in each proof are showing that H is character-

For a given group G and a homomorphism ϕ: G → G × G, we construct groups ℱϕ(G), 𝒯ϕ(G), and 𝒱ϕ(G) that blend Thompson's groups F, T, and V with G, respectively. Furthermore, we describe the lattice of normal subgroups of the groups ℱΔ(G), where Δ: G → G × G is the diagonal homomorphism, Δ(g) = (g, g).

In this paper, we discuss the sublattices of intuitionistic fuzzy subgroups of a given group and some special sublattices corresponding to it. We prove that a class of intuitionistic fuzzy normal subgroups constitutes a modular of the lattice of intuitionistic fuzzy normal subgroups.

A subgroup H of a group G is said to be nearly normal in G if it has a finite index in its normal closure H G.T he setnn(G) of nearly normal subgroups of G is a sublattice of the lattice of all subgroups of G. Isomorphisms between lattices of nearly normal subgroups of FC -soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated. Moreover, it is proved that if G is a supersoluble group and ¯ G is an FC -soluble group such that the lattices nn(G )a ndnn( ¯ G )a re isomorphic, then also ¯ G is supersoluble.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

We introduce and study the lattice of normal subgroups of a group G that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of G, see [Kaplan G., Levy D., Solitary subgroups, Comm. Algebra, 2009, 37(6), 1873–1883]. A precise description of this lattice is given for some particular classes of finite groups.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

On diagrams for abelian groups

In the paper the concept of the similarity of homogeneously decomposable groups is introduced. One investigates the cases in which almost isomorphic groups are similar or isomorphic.

One way of expressing the self-duality $A\cong {\rm Hom}(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe p-group examples.