Direct Product Of Cyclic Groups Research Articles

Factoring Subgroups and Factor Groups of the Unit Group Modulo n

Summary We introduce several ways of producing subgroups of U(n), the group of units of Zn . Our results find the structure of these various subgroups in terms of external direct product of cyclic groups. We then use our classifications to give descriptions of elements of U(n) that form a subgroup of U(n) with a desired cyclic group decomposition. This includes a description of elements that form a Sylow p-subgroup.

Groups of units of Zp[x] modulo f(x)

The set $\mathbb{Z}_p[x]$ consists of all polynomials with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. If a polynomial $f(x)$ is irreducible over $\mathbb{Z}_p$ then $\frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}$ is a field. If $f(x)$ is a reducible polynomial, then every non-zero element in $\frac{\mathbb{Z}_p[x] }{\langle f(x) \rangle}$ is either a zero-divisor or a unit. If we exclude the zero-divisors and zero, we have a finite Abelian group under multiplication denoted by $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $. Since every finite Abelian group is a direct product of cyclic groups of prime-power order, we can find the isomorphism class for $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $. We investigate the structure of $U \Big( \frac{\mathbb{Z}_p[x] }{ \langle f(x) \rangle}\Big) $ for a prime $p$ and various $f(x)$. We conclude with some result on the structure of a certain family of subgroups of $U \Big( \frac{\mathbb{Z}_p[x]}{\langle f(x)\rangle} \Big)$.

The structure of centralizers in the finite symmetric inverse semigroup

For an integer let be the symmetric inverse semigroup of partial injective transformations on an n-element set. For denote by and respectively, the first and second centralizer of λ in We determine the structure of and in terms of Green’s relations, including the partial order of -classes, and express as a direct product of cyclic groups with zero adjoined and a monogenic monoid. For each individual Green relation we determine such that in is inherited from and λ such that all Green relations in are inherited from We also provide a representation of the partial order of -classes in as a known lattice, and describe such that which gives a class of maximal commutative subsemigroups of

Finite groups which are c-absolute central factor group and the union of Ac-centralizers

Let [Formula: see text] be a group and [Formula: see text] be the [Formula: see text]-absolute center of [Formula: see text], that is, the set of all elements of [Formula: see text] fixed by all class preserving automorphisms of [Formula: see text]. In this paper, we classify all finite groups [Formula: see text], whose [Formula: see text]-absolute central factors are isomorphic to the direct product of cyclic groups, [Formula: see text] and [Formula: see text]. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of [Formula: see text] for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

Molecular symmetry: Why permutation-inversion (PI) groups don’t render the point groups obsolete

The analysis of spectra of molecules with internal large-amplitude motions (LAMs) requires molecular symmetry (MS) groups that are larger than and significantly different from the more familiar point groups. MS groups are described often by the permutation-inversion (PI) group method. It is shown that point groups still can and should play a significant role together with the PI groups for a class of molecules with internal rotors. In molecules of this class, several simple internal rotors are attached to a rigid molecular frame. The PI groups for this class are semidirect products like H ^ F, where the invariant subgroup H is a direct product of cyclic groups and F is a point group. This result is used to derive meaningful labels for MS groups, and to derive correlation tables between MS groups and point groups. MS groups of this class have many parallels to space groups of crystalline solids.

The Lausch group simplified: A direct product decomposition

The Lausch group of a Lockett class of groups is decomposed into a direct product of cyclic groups under very general hypotheses applicable to many Fitting pairs defined in the literature. The Fitting pairs are shown to exhibit strong independence properties. The results apply to the class of all solvable groups, and, therefore, to all Lockett classes satisfying the Lockett conjecture.

A classification of finite quantum kinematics

Quantum mechanics in Hilbert spaces of finite dimension N is reviewed from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite Abelian groups. This yields also a classification of finite Weyl-Heisenberg groups and the corresponding finite quantum kinematics. Simple number theory gets involved through the fundamental theorem describing all finite discrete Abelian groups of order N as direct products of cyclic groups, whose orders are powers of not necessarily distinct primes contained in the prime decomposition of N. The representation theoretic approach is further compared with the algebraic approach, where the basic object is the corresponding operator algebra. The consideration of fine gradings of this associative algebra then brings a fresh look on the relation between the mathematical formalism and physical realizations of finite quantum systems.

The complexity of intersecting finite automata having few final states

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be $${\oplus}$$źL-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata. We further consider idempotent commutative automata and (Abelian, mainly) group automata with one, two, or three final states over a singleton or larger alphabet, elucidating (under the usual hypotheses on complexity classes) the complexity of the intersection nonemptiness and related problems in each case.

Isomorphisms of Direct Products of Cyclic Groups of Prime Power Order

Summary In this paper we formalized some theorems concerning the cyclic groups of prime power order. We formalize that every commutative cyclic group of prime power order is isomorphic to a direct product of family of cyclic groups [1], [18].

Integral Cayley Graphs Defined by Greatest Common Divisors

An undirected graph is called integral, if all of its eigenvalues are integers. Let $\Gamma =Z_{m_1}\otimes \ldots \otimes Z_{m_r}$ be an abelian group represented as the direct product of cyclic groups $Z_{m_i}$ of order $m_i$ such that all greatest common divisors $\gcd(m_i,m_j)\leq 2$ for $i\neq j$. We prove that a Cayley graph $Cay(\Gamma,S)$ over $\Gamma$ is integral, if and only if $S\subseteq \Gamma$ belongs to the the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. It is also shown that every $S\in B(\Gamma)$ can be characterized by greatest common divisors.

Identities of the soluble product of abelian groups

We consider the product G of abelian groups in the variety \(\mathfrak{A}^n \) of soluble groups of length at most n. Provided that the abelian factors are decomposable into direct products of cyclic groups, we find necessary and sufficient conditions for G to generate the variety \(\mathfrak{A}^n \).

Universal diagram groups with identical Poincaré series

For a diagram group G, the first derived quotient G(1)/G(2) is always free abelian (as proved by M. Sapir and V. Guba). However the second derived quotient G(2)/G(3) may contain torsion. In fact, we show that for any finite or countably infinite direct product of cyclic groups A, there is a diagram group with second derived quotient A. We use that to construct families with the properties of the title.

Linear and sublinear time algorithms for the basis of abelian groups

It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ≅ G 1 × G 2 × ⋯ × G t , where each G i is a cyclic group of order p j for some prime p and integer j ≥ 1 . If a i generates the cyclic group of G i , i = 1 , 2 , … , t , then the elements a 1 , a 2 , … , a t are called a basis of G . We show a randomized algorithm such that given a set of generators M = { x 1 , … , x k } for an abelian group G and the prime factorization of order ord ( x i ) ( i = 1 , … , k ) , it computes a basis of G in O ( | M | ( log n ) 2 + ∑ i = 1 t n i p i n i / 2 ) time, where n = | G | has prime factorization p 1 n 1 p 2 n 2 ⋯ p t n t (which is not a part of input). This generalizes Buchmann and Schmidt’s algorithm that takes O ( | M | | G | ) time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O ( n ) time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group.

Orthogonal wavelets on direct products of cyclic groups

We describe a method for constructing compactly supported orthogonal wavelets on a locally compact Abelian group G which is the weak direct product of a countable set of cyclic groups of pth order. For all integers p, n ≥ 2, we establish necessary and sufficient conditions under which the solutions of the corresponding scaling equations with pn numerical coefficients generate multiresolution analyses in L2(G). It is noted that the coefficients of these scaling equations can be calculated from the given values of pn parameters using the discrete Vilenkin-Chrestenson transform. Besides, we obtain conditions under which a compactly supported solution of the scaling equation in L2(G) is stable and has a linearly independent system of “integer” shifts. We present several examples illustrating these results.

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.

Manifolds with finite cyclic fundamental groups and codimension 2 fibrators

A closed connected n-manifold N is called a codimension 2 fibrator (codimension 2 orientable fibrator, respectively) if each proper map p:M→B on an (orientable, respectively) (n+2)-manifold M each fiber of which is shape equivalent to N is an approximate fibration. Let r be a nonnegative integer and let N be a closed n-manifold whose fundamental group is isomorphic to H 1×H 2 , where H 1 is a group whose order is odd and H 2 is a finite direct product of cyclic groups of order 2 r . Let q 1:N 1→N be the covering associated with H 1 . The main purpose of this paper shows that if N 1 is a codimension 2 orientable fibrator, then N is a codimension 2 fibrator.

Students Ask the Darnedest Things: A Result in Elementary Group Theory

Introduction In most elementary group theory courses, students are introduced to the Fundamental Theorem of Finite Abelian Groups (FT), which states that every finite abelian group is isomorphic to exactly one direct product of cyclic groups of prime power order. (These cyclic groups are called invariant factors.) As every cyclic group of order k is isomorphic to Zk, the additive group of integers mod k, FT asserts that every abelian group of order 12 is isomorphic either to Z4 X Z3 or to Z2 X Z2 X Z3. (I'll use the common abbreviation Zk to denote Z/kZ, with apologies to my fellow number theorists, who usually use this notation to denote the k-adic integers.) A standard exercise is to determine whether a given abelian group G of order 12 is isomorphic to Z4 X Z3 or Z2 X Z2 x Z3 by computing the orders of the elements of G. For instance, if G contains an element of order 12 (or 4, for that matter), it must be isomorphic to the cyclic group Z4 X Z3, and if G has more than 1 element of order 2, then it is isomorphic to Z2 X Z2 X Z3. Notice, however that every abelian group of order 12 contains exactly two elements of order 3. An important concept reinforced through such an exercise is that isomorphisms preserve order. Specifically, if f: G -> G' is a group isomorphism, then g and f(g) have the same order in their respective groups. Thus if two finite groups G and G' are isomorphic, they must have identical order structure (the same number of elements of each order). This article examines the converse question, posed by a student in a group theory course:

(Spontaneously broken) Abelian Chern-Simons theories

A detailed analysis of Chem-Simons (CS) theories in which a compact Abelian direct product gauge group U(1) k is spontaneously broken down to a direct product of cyclic groups H ⋍ Z N (1) × … x Z N (k) is presented. The spectrum features global H charges, vortices carrying magnetic flux labeled by the elements of H and dyonic combinations. Due to the Aharonov-Bohm effect these particles exhibit topological interactions. The remnant of the U(1) k CS term in the discrete H gauge theory describing the effective long distance physics of such a model is shown to be a 3-cocycle for H governing the non-trivial topological interactions for the magnetic fluxes implied by the U(1) k CS term. It is noted that there are in general three types of 3-cocycles for a finite Abelian gauge group H: one type describes topological interactions between vortices carrying flux with respect to the same cyclic group in the direct product H, another type gives rise to topological interactions among vortices carrying flux with respect to two different cyclic factors of H and a third type leading to topological interactions between vortices carrying flux with respect to three different cyclic factors. Among other things, it is demonstrated that only the first two types can be obtained from a spontaneously broken U(1) k CS theory. The 3-cocycles that cannot be reached in this way turn out to be the most interesting. They render the theory non-Abelian and in general lead to dualities with planar theories with a non-Abelian finite gauge group. In particular, the CS theory with finite gauge group H ⋍ Z 2 × Z 2 × Z 2 defined by such a 3-cocycle is shown to be dual to the planar discrete D 4 gauge theory with D 4 the dihedral group of order 8.

Symmorphic space groups of finite crystal lattices

The symmorphic space group of a finite n-dimensional crystal lattice is studied and its factorization is presented. This lattice is determined by a direct product of cyclic groups and then its translation, point and space groups are defined as modular images of corresponding ones for a lattice of infinite extent. A factorization of holohedries (of an infinite lattice) is used in order to present a symmorphic finite space group as a direct product. As an example, and as a very special case, a space group in the case of the hypercubic lattice is studied and its irreducible representations are investigated. The results obtained suggest that for vectors lying on a surface of the first Brillouin zone (i.e. for which at least one coordinate is equal to 0 or 1/2) an additional index describing their symmetry properties should be introduced. This enables the authors to make a more detailed classification of states (energy levels). On the other hand, finite symmorphic space groups can be used within, e.g., the so-called finite-lattice approach.

Fuzzy subgroups: Some characterizations

The notion of a fuzzy set was first introduced in 1965 by Zadeh [12]. Since then the theory of fuzzy sets has gone through remarkably rapid strides with well over 4000 papers by now (see, for example, the bibliography [3]), several textbooks (for example, [2 and 6]), and an international journal solely devoted to it [S]. The theory has found wideranging applications in such diverse fields as automata, control theory, decision theory, and social behavior pattern studies, to name just a few. For an interesting, expository account see Rosenfeld [lo], and also the book review [7]. In [9], Rosenfeld introduced the notion of a fuzzy group and showed that many concepts of group theory can be extended in an elementary manner to develop the theory of fuzzy groups. In particular, he characterized all the fuzzy subgroups of cyclic groups of prime order [9, Proposition 5.101. Fuzzy groups were further investigated by Das in [1] where he characterized all the fuzzy subgroups of finite cyclic groups [l, Theorem 4.21. In this connection, he introduced the notion of “level subgroups” of a fuzzy subgroup which is based on the notion of a “level subset” introduced earlier by Zadeh. For a finite group Das showed that these level subgroups of a fuzzy subgroup form a chain [ 1, Corollary 3.11 and also constructed a fuzzy subgroup whose chain of level subgroups is maximal in the following special cases when the group is: (i) a direct product of cyclic groups of prime orders, (ii) abelian [l, Theorem 3.41. We analyze the level subgroups of a fuzzy subgroup in more detail and investigate whether the family of level subgroups of a fuzzy subgroup deter-