Abelian Permutation Group Research Articles

On 2-closed abelian permutation groups

A permutation group is said to be 2-closed if no group H such that has the same orbits on as G. A simple and efficient inductive criterion for the 2-closedness is established for abelian permutation groups with cyclic transitive constituents.

Abelian permutation groups with graphical representations

In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur’s classical terminology, what we provide is characterizations of the classes of 2-closed and 2^*-closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.

Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group

Abstract We present a construction of all finite indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case; in particular, we show they are regular abelian groups.

On the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation of prime-power size

We develop a method to construct all the indecomposable involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the indecomposable ones having abelian permutation group and cardinality pq (where p and q are prime numbers not necessarily distinct).

2-closed abelian permutation groups

In this paper we demonstrate that the result by Zelikovskij concerning Königs problem for abelian permutation groups, reported in a recent survey, is false. We propose in this place two results on 2-closed abelian permutation groups which concern the same topic in a more general setting.

Unique maximal rings of functions

ABSTRACTFor several classes of groups G, we characterize when the near-ring M0(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M0(G) are commutative.

On the Diameters of Commuting Graphs of Permutational Wreath Products

Let G be a group and let Z(G) be the center of G. The commuting graph of the group G is an undirected graph Γ(G) with the vertex set G \Z (G) such that two vertices x,y are adjacent if and only if xy = yx. We study the commuting graphs of permutational wreath products H o G, where G is a transitive permutation group acting on X (the top group of the wreath product) and (H,Y ) is an Abelian permutation group acting on Y.

Symmetry groups of boolean functions

We prove that every abelian permutation group, but known exceptions, is the symmetry group of a boolean function. This solves the problem posed in the book by Clote and Kranakis. In fact, our result is proved for a larger class of permutation groups, namely, for all subgroups of direct sums of regular permutation groups.

The Parallel Complexity of Graph Canonization Under Abelian Group Action

We study the problem of computing canonical forms for graphs and hypergraphs under Abelian group action and show tight complexity bounds. Our approach is algebraic. We transform the problem of computing canonical forms for graphs to the problem of computing canonical forms for associated algebraic structures, and we develop parallel algorithms for these associated problems. 1. In our first result we show that the problem of computing canonical labelings for hypergraphs of color class size 2 is logspace Turing equivalent to solving a system of linear equations over the field \(\mathbb {F} _{2}\). This implies a deterministic NC2 algorithm for the problem. 2. Similarly, we show that the problem of canonical labeling graphs and hypergraphs under arbitrary Abelian permutation group action is fairly well characterized by the problem of computing the integer determinant. In particular, this yields deterministic NC3 and randomized NC2 algorithms for the problem.

Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes

In this paper we classify the complexity of several problems based on Abelian permutation groups and linear congruences using logspace counting classes. The problems we consider were defined by McKenzie & Cook (1987). Central to our study is the problem LCON: given as input (A, b, q), where $$A \in {\mathbb{Z}}^{m \times n}$$and b $$\in {\mathbb{Z}}^m$$, the problem is to determine if Ax = b is a feasible system of linear equations over $${\mathbb{Z}}_q$$. We assume that q is given by its prime factorization $$q = p^{e_1}_{1} p^{e_2}_{2} \cdot \cdot \cdot p^{e_k}_{k}$$, such that each $$p^{e_i}_i$$is tiny (i.e. given in unary). We give a randomized NC2 algorithm for LCON. More precisely, LCON is in the nonuniform class LGapL/poly. As LCON is hard for LGapL we get a fairly tight characterization of LCON in terms of logspace counting classes. We derive the same upper bound for computing a basis for the nullspace of a linear map from $${\mathbb{Z}}^n_q$$to $${\mathbb{Z}}^m_q$$. A number of Abelian permutation group problems studied in McKenzie & Cook (1987) turn out to be logspace Turing equivalent to these linear-algebraic problems. Consequently, the upper and lower bounds also carry over to these problems.

Simple current symmetries in RCFT

The question ''Which abelian permutation groups arise as group of simple currents in Rational Conformal Field Theory?'' is investigated using the formalism of weighted permutation actions. After a review of the relevant properties of simple current symmetries, the general theory of WPA-s and admissibility conditions are described, and classification results are illustrated by a couple of examples.

Rooted wreath products

We introduce “rooted valuation products” and use them to construct universal Abelian lattice-ordered groups (with prescribed set of components) from the more classical theory of H. Hahn. The wreath product construction of W.C. Holland and S.H. McCleary generalised the Abelian (lattice-ordered) permutation group ideas to give universals for transitive (ℓ-)permutation groups with prescribed set of primitive components. In the case of (not necessarily transitive) sublattice subgroups of order-preserving permutations of totally ordered sets, the set of natural congruences forms a root system. We generalise the rooted valuation product construction to the permutation case when all natural primitive components are regularly obtained; we analogously obtain universals for these permutation groups (for a prescribed set of natural primitive components) which we call “rooted wreath products.” We identify the rooted valuation product with an appropriate subgroup of the corresponding rooted wreath product. The maximal Abelian group actions on the ordered real line were characterised in by R. Winkler, and their digital representations were consequently obtained. We use the rooted wreath product construction to get a more general result, and deduce Winkler's characterisation as a consequence.

On the abelian inner permutation groups of loops

Let Q be a loop such that the inner permutation group I(Q) of Q is finite and abelian. The aim of the present note is to show that Q is then centrally nilpotent and no non-trivial primary component of I(Q) is cyclic. Notice that this is a generalization of [5, 2.1,2.4] and [7, 6.2,6.4], where the mentioned results are proved for finite loops.

On the computational complexity of the abelian permutation group structure, membership and intersection problems

Algorithms on computations on abelian permutation groups are presented here. An algorithm for computing the complete structure, algorithms for membership-inclusion testing and an algorithm for computing the intersection of abelian permutation groups are given. Their worst-case time complexity is a polynomial of degree 4 in terms of n, the number of points moved by the group. The upper bounds on the running time of the algorithms shown here improve the bounds on the above problems cited in the literature.

The Parallel Complexity of Abelian Permutation Group Problems

We classify Abelian permutation group problems with respect to their parallel complexity. For such groups specified by generating permutations we show that testing membership, computing order and testing isomorphism are $NC^1 $-equivalent to (and therefore have essentially the same parallel complexity as) determining solvability of a system of linear equations modulo a product of small prime powers; we show that intersecting two such groups is $NC^1 $-equivalent to computing setwise stabilizers; we show that each of these problems is $NC^1 $-reducible to the problem of computing a generator-relator presentation. Then we prove that the aforementioned problems belong to $NC^3 $, thus identifying several natural set recognition problems in $NC$ which may lie outside $NC^2 $. Finally we prove that $NC^4 $ contains the problem of computing the cyclic decomposition of an Abelian permutation group. Background results include an $NC^1 $ solution to the problem of computing the product of n integers modulo a $\lce...

Monte Carlo circuits for the abelian permutation group intersection problem

We show that the problem of computing a basis for an abelian transitive permutation group is in N C k and also we show that the problem of computing a basis for an abelian permutation group and the problem of computing the intersection of two abelian groups acting on n points, can be solved in depth (log n)k on a Monte Carlo Boolean circuit of polynomial size. Moreover the latter two problems are shown to be in N C k in the restricted case of bounded number of generators.

Relative difference sets and quasiregular collineation groups

A permutation group is called quasiregular if it acts regularly on each of its orbits. In particular, any abelian permutation group is quasiregular. In [3], the following result was proved. THEOREM 1.1. Let F be a quasiregular collineation group acting on a finite projective planezr, of order n. If[ F t > 89 ~ -k n + 1), and if t denotes the number of point (or line) orbits of and if F denotes the fixed sub- structure of[', then one of the following must hold. F =n2-kn+ 1, t= 1, F= ~. 1 = n ~, t = 3, F is an incident point-line pair (A, m). [' = n ~, t = n + 2, F is either a line and all its points, or

Automorphisms of functions in abelian permutation groups

Let Ω = H1⊕…⊕Hn be an abelian group of permutations of a finite non-empty set S. If Hi is generated by φi, let sφi(α) denote the length of the cycle of φi containing α. For any function f on S, let A(f,Ω) = (φ ∈ Ω|fφ = f). In Theorem 2 we show that, if for every i ≠ j and α ∈ S, Sφi(α) and Sφj(α) are relatively prime, then A(f, Ω) = A(f, H1)⊕…⊕A(f, Hn) for all f, while in Theorem 3 we prove the natural converse.