scholarly journals A family of difference sets in non-cyclic groups

1973 ◽  
Vol 15 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Robert L McFarland
Keyword(s):  
Difference Sets ◽  
Cyclic Groups

scholarly journals Nonexistence of certain supplementary difference sets

1975 ◽  
Vol 13 (3) ◽  
pp. 343-348 ◽  
Author(s):  
Nicholas Wormald
Keyword(s):  
Difference Sets ◽  
Cyclic Groups ◽  
Supplementary Difference Sets ◽  
Supplementary Difference ◽  
Index 2

This paper finds restrictions on the parameters of supplementary difference sets in any group G with a subgroup of index 2, which therefore includes all cyclic groups of even orders. As a corollary to the main theorem, we have that if S1, …, Sr are r − {2v, k1, …, kr; 2λ} supplementary difference sets in such a group, then not all of v, k1, …, kr, λ are odd; also is the sum of r squares.

scholarly journals Inequivalence of Difference Sets: On a Remark of Baumert

10.37236/2277 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Padraig Ó Catháin
Keyword(s):  
Difference Sets ◽  
Cyclic Groups ◽  
Cyclic Difference Sets

An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic $(4t-1,2t-1,t-1)$ difference sets are inequivalent, apart from a small number of exceptions with $t< 8$. We are not aware of a proof of this statement in the literature.Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases.

On the minimal varieties of PI-algebras graded by finite cyclic groups

2021 ◽  
pp. 1-25
Author(s):  
Marcos Antônio da Silva Pinto ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Cyclic Groups

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

scholarly journals Graphs of vectorial plateaued functions as difference sets

2021 ◽  
Vol 71 ◽  
pp. 101795
Author(s):  
Ayça Çeşmelioğlu ◽  
Oktay Olmez
Keyword(s):  
Difference Sets ◽  
Plateaued Functions

Divisible Semiplanes, Arcs, and Relative Difference Sets

1987 ◽  
Vol 39 (4) ◽  
pp. 1001-1024 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Projective Plane ◽  
Difference Sets ◽  
Difference Set ◽  
Relative Difference ◽  
List Type ◽  
Relative Difference Set ◽  
Large Collection ◽  
Baer Subplane ◽  
Relative Difference Sets ◽  
Definition Of

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

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