Abelian actions on compact nonorientable Riemann surfaces

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

Graceful labeling of power graph of group Z2k−1 × Z4

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let [Formula: see text] be a finite abelian 2-group of order [Formula: see text] where [Formula: see text]. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on [Formula: see text].

Evaluation of effectiveness of chosen-plaintext attacks on the Rao-Nam cryptosystem over a finite Abelian group

The Rao-Nam cryptosystem is a symmetric version of the McEliece code-based cryptosystem proposed to get rid of the shortcomings inherent in the first symmetric code-based encryption schemes. Almost immediately after the publication of this cryptosystem, attacks on it based on selected plaintexts appeared, which led to the emergence of various improvements and modifications of the original cryptosystem. The secret key in the traditional Rao-Nam scheme is a certain Boolean matrix and a set of binary vectors used to generate distortions during encryption. Such vectors must have different syndromes, that is, be different modulo of the code generated by the rows of the specified matrix. The original work of Rao and Nam considered two methods of forming the set of these vectors, the first of which consists in using predetermined vectors of sufficiently large weight, and the second is random selection of these vectors according to the equiprobable scheme. It is known that the first option does not provide the proper security of the Rao – Nam cryptosystem (due to the small number and simple structure of these vectors), but the second option is more meaningful and requires additional research. The purpose of this paper is to obtain estimates of the effectiveness (time complexity for a given upper bound of the error probability) of attacks on a cryptosystem, which generalizes the traditional Rao – Nam scheme to the case of a finite Abelian group (note that the need to study such versions of the Rao – Nam cryptosystem is due to their consideration in recent publications). Two attacks, based on selected plaintext, are presented. The first of them is not mentioned in the works known to the authors of this article and, under certain well-defined conditions, it allows recovering the secret key of the cryptosystem with quadratic complexity. The second attack is a generalized and simplified version of the well-known Struik-van Tilburg attack. It is shown that the complexity of this attack depends on the power of the stabilizer of the set of vectors, which forms the second part of the key, in the translation group of the Abelian group, over which the Rao – Nam cryptosystem is considered. In this paper, a bound is obtained for the probability of triviality of the stabilizer under the condition of random choice of this set. From the obtained bound, it follows that Struik-van Tilburg attack is, on average, noticeably more efficient than the worst case considered earlier.

Additive bases via Fourier analysis

Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Groups containing locally maximal product-free sets of size 4

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

The probability of an automorphism of an abelian group fixing a group element

In this paper, we consider the probability of an automorphism of a finite abelian group fixing a group element. Explicit computations are made to find the fusion classes of a finite abelian groups. The probability of an automorphism fixing a group element is obtained in terms of fusion classes. We also compute the bounds of the probability for some particular cases.

EXISTENCE CONDITIONS FOR k-BARYCENTRIC OLSON CONSTANT

Let (G, +) be a finite abelian group and 3 ≤ k ≤ |G| a positive integer. The k-barycentric Olson constant denoted by BO(k, G) is defined as the smallest integer ℓ such that each set A of G with |A| = ℓ contains a subset with k elements {a1, . . . , ak} satisfying a1 + · · · + ak = kaj for some 1 ≤ j ≤ k. We establish some general conditions on G assuring the existence of BO(k, G) for each 3 ≤ k ≤ |G|. In particular, from our results we can derive the existence conditions for cyclic groups and for elementary p-groups p ≥ 3. We give a special treatment over the existence condition for the elementary 2-groups.