Search results for "group theory"
showing 10 items of 703 documents
A Classification of all Symmetric Block Designs of Order Nine with an Automorphism of Order Six
2006
We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 301–312, 2006
Thin bases of order h
2003
Abstract A subset A⊆ N 0 is called a basis of order h if every positive integer can be represented as a sum of h members of A . Thin bases of order h will be constructed in this paper, for each h ⩾2, where the value of lim sup A(n)/ n h is smaller than that of thin bases known so far. In the most important case h =2 it is shown that for the considered class of bases (which generalizes an ansatz of Stohr) the result is best possible up to an e >0.
Invariant characters and coprime actions on finite nilpotent groups
2000
Suppose that a group A acts via automorphisms on a nilpotent group G having coprime order. Given an A-invariant character \(\chi \in {\rm Irr}(G)\), we show that the A-primitive irreducible characters that induce \(\chi \) from an A-invariant subgroup of G all have equal degree. We use this result to obtain some information about the characters of groups of p-length 1.
Verbal sets and cyclic coverings
2010
Abstract We consider groups G such that the set of all values of a fixed word w in G is covered by a finite set of cyclic subgroups. Fernandez-Alcober and Shumyatsky studied such groups in the case when w is the word [ x 1 , x 2 ] , and proved that in this case the corresponding verbal subgroup G ′ is either cyclic or finite. Answering a question asked by them, we show that this is far from being the general rule. However, we prove a weaker form of their result in the case when w is either a lower commutator word or a non-commutator word, showing that in the given hypothesis the verbal subgroup w ( G ) must be finite-by-cyclic. Even this weaker conclusion is not universally valid: it fails …
Graph connectivity and monadic NP
2002
Ehrenfeucht games are a useful tool in proving that certain properties of finite structures are not expressible by formulas of a certain type. In this paper a new method is introduced that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy. As an application it is shown that graph connectivity cannot be expressed by existential second-order formulas, where the second-order quantification is restricted to unary relations (monadic NP), even, in the presence of a built-in linear order. As a second application it is stated, that, on the other hand, the presence of a linear order increases the power of monadi…
Degree of monotonicity in aggregation process
2010
In this paper we introduce a fuzzy order relation notion in the description of aggregation process. Namely, we use the fuzzy order relation to define the degree of monotonicity, which is equal to 1 for a monotone function with respect to a crisp order relation. In that case, integration of fuzzy order relation allows us to generalize the notion of monotonicity and we try to investigate the benefits of using fuzzy relations instead of a crisp relation. Further we illustrate this definition by examples and study the properties of aggregation functions which have a certain degree of monotonicity.
Sensitivity Versus Certificate Complexity of Boolean Functions
2016
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that the certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $$bsf \le Cf \le 2^{sf-1} sf - sf-1$$. In this work, we give a better upper bound of $$bsf \le Cf \le \max \left 2^{sf-1}\left sf-\frac{1}{3}\right , sf\right $…
Universal Lyndon Words
2014
A word w over an alphabet Σ is a Lyndon word if there exists an order defined on Σ for which w is lexicographically smaller than all of its conjugates (other than itself). We introduce and study universal Lyndon words, which are words over an n-letter alphabet that have length n! and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every n and exhibit combinatorial and structural properties of these words. We then define particular prefix codes, which we call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in bijection with the set of the shortest unrepeated prefixes of the conjugates of a universal Lyndon word. This allows us t…
McKay natural correspondences on characters
2014
Let [math] be a finite group, let [math] be an odd prime, and let [math] . If [math] , then there is a canonical correspondence between the irreducible complex characters of [math] of degree not divisible by [math] belonging to the principal block of [math] and the linear characters of [math] . As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow [math] -subgroup or a [math] -decomposable Sylow normalizer.
On the supersoluble hypercentre of a finite group
2016
[EN] We give some sufficient conditions for a normal p-subgroup P of a finite group G to have every G-chief factor below it cyclic. The S-permutability of some p-subgroups of O^p(G)plays an important role. Some known results can be reproved and some others appear as corollaries of our main theorems.