In 1998, Khodkar showed that the minimal critical set in the Latin square corresponding to the elementary abelian 2-group of order 16 is of size at most 124. Since the paper was published, improved methods for solving integer programming problems have been developed. Here we give an example of a critical set of size 121 in this Latin square, found through such methods. We also give a new upper bound on the size of critical sets of minimal size for the elementary abelian 2-group of order $2^n$: $4^{n}-3^{n}+4-2^{n}-2^{n-2}$. We speculate about possible lower bounds for this value, given some other results for the elementary abelian 2-groups of orders 32 and 64. An example of a critical set of size 29 in the Latin square corresponding to the elementary abelian 3-group of order 9 is given, and it is shown that any such critical set must be of size at least 24, improving the bound of 21 given by Donovan, Cooper, Nott and Seberry.
Tarnauceanu [Archiv der Mathematik, 102 (1), (2014), 11--14] gave a characterisation of elementary abelian $2$-groups in terms of their maximal sum-free sets. His theorem states that a finite group $G$ is an elementary abelian $2$-group if and only if the set of maximal sum-free sets coincides with the set of complements of the maximal subgroups. A corollary is that the number of maximal sum-free sets in an elementary abelian $2$-group of finite rank $n$ is $2^n-1$. Regretfully, we show here that the theorem is wrong. We then prove a correct version of the theorem from which the desired coroll…
Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified. A nature question is to classify regular Cayley maps of elementary abelian $p$-groups $Z_p^n$. In this paper, a complete classification of regular Cayley maps of $Z_p^n$ is given and moreover, the number of these maps and their genera are enumerated.
For every prime $p > 2$ we exhibit a Cayley graph of $\mathbb{Z}_p^{2p+3}$ which is not a CI-graph. This proves that an elementary Abelian $p$-group of rank greater than or equal to $2p+3$ is not a CI-group. The proof is elementary and uses only multivariate polynomials and basic tools of linear algebra. Moreover, we apply our technique to give a uniform explanation for the recent works concerning the bound.
A skew-morphism of a finite group $G$ is a permutation $σ$ on $G$ fixing the identity element, and for which there exists an integer function $π$ on $G$ such that $σ(xy)=σ(x)σ^{π(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $σ$ of $G$, the product of $\langle σ\rangle$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $σ$. In this paper, the skew-product groups of skew-morphisms of finite elementary abelian $p$-groups are investigated. Some properties, characterizations and constructions about that are obt…
Let A and B be subsets of an elementary abelian 2-group G, none of which are contained in a coset of a proper subgroup. Extending onto potentially distinct summands a result of Hennecart and Plagne, we show that if |A+B|<|A|+|B|, then either A+B=G, or the complement of A+B in G is contained in a coset of a subgroup of index at least 8, whence |A+B| is at least 7/8 |G|. We indicate conditions for the containment to be strict, and establish a refinement in the case where the sizes of A and B differ significantly.
In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of rank $2$.
Let $\p_N \colon \tX \to X$ be a regular covering projection of connected graphs with the group of covering transformations isomorphic to $N$. If $N$ is an elementary abelian $p$-group, then the projection $\p_N$ is called $p$-elementary abelian. The projection $\p_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$ lifts along $\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection $\p_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition $\p_N = \p \circ \p_M$ of two (nontrivial) regular cover…
We determine the BP-module structure, mod higher filtration, of the main part of the BP-homology of elementary abelian 2-groups. The action is related to symmetric polynomials and to Dickson invariants.
We classify the orientably regular maps which are elementary abelian regular branched coverings of Platonic maps M, in the case where the covering group and the rotation group G of M have coprime orders. The method involves studying the representations of G on certain homology groups of the sphere, punctured at the branch-points. We give a complete classification for branching over faces (or, dually, vertices) of M, and outline how the method extends to other branching patterns.
In this paper, we show that the group $\mathbb{Z}_p^5$ is a DCI-group for any odd prime $p,$ that is, two Cayley digraphs Cay$(\mathbb{Z}_p^5,S)$ and Cay$(\mathbb{Z}_p^5,T)$ are isomorphic if and only if $S=T^φ$ for some automorphism $φ$ of the group $\mathbb{Z}_p^5$.
A finite group $R$ is a DCI-group if, whenever $S$ and $T$ are subsets of $R$ with the Cayley graphs ${\rm Cay}(R,S)$ and ${\rm Cay}(R,T)$ isomorphic, there exists an automorphism $φ$ of $R$ with $S^φ=T$. Elementary abelian groups of order $p^4$ or smaller are known to be DCI-groups, while those of sufficiently large rank are known not to be DCI-groups. The only published proof that elementary abelian groups of order $p^4$ are DCI-groups uses Schur rings and does not work for $p=2$ (which has been separately proven using computers). This paper provides a simpler proof that works for all primes…
The existence of an extremal self-dual binary linear code C of length 72 is a long-standing open problem. We continue the investigation of its automorphism group: looking at the combination of the subcodes fixed by different involutions and doing a computer calculation with Magma, we prove that Aut(C) is not isomorphic to the elementary abelian group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5.
In this paper we determine a class of critical sets in the abelian {2--group} that may be obtained from a greedy algorithm. These new critical sets are all 2--critical (each entry intersects an intercalate, a trade of size 4) and completes in a top down manner.
In this paper, we study self-complementary distance-regular Cayley graphs over abelian groups. We prove that if a regular graph is self-complementary distance-regular, then it is self-complementary strongly regular. We also deal with self-complementary strongly regular Cayley graphs over abelian groups and give an example of a self-complementary strongly regular Cayley graph over a non-elementary abelian group.
In this paper, we consider regular automorphism groups of graphs in the RT$2$ family and the Davis-Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non-abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT$2$ family and Davis-Xiang family and their amorphic abelian Cayley schemes to produce amorphic non-abelian Cayley schemes.
A difference set is said to have classical parameters if $ (v,k, λ) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian difference sets with $d=4$. The only known examples of such difference sets correspond to the projective geometries $PG(3,q)$. We consider an arbitrary difference set with the parameters of $PG(3,q)$ in an abelian group and establish constraints on its structure. In particular, we discern embedded substructures.
The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the Möbius function is given for the general abelian group, while in many special cases, including the case of elementary abelian groups, solved earlier by Li and Wan, it has a compact form. The proof involves two different Möbius transforms, on positive integers and on set partitions.
Partial difference sets with parameters $(v,k,λ,μ)=(v, (v-1)/2, (v-5)/4, (v-1)/4)$ are called Paley type partial difference sets. In this note we prove that if there exists a Paley type partial difference set in an abelian group $G$ of an order not a prime power, then $|G|=n^4$ or $9n^4$, where $n>1$ is an odd integer. In 2010, Polhill \cite{Polhill} constructed Paley type partial difference sets in abelian groups with those orders. Thus, combining with the constructions of Polhill and the classical Paley construction using non-zero squares of a finite field, we completely answer the following…
A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a $CI$-Schur ring. By using this condition we reprove in a short way known results on the $CI$-property for decomposable Schur rings over an elementary abelian group of rank at most $5$.
Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.
The Cayley-Dickson loop Q_n is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We establish that the inner mapping group Inn(Q_n) is an elementary abelian 2-group of order 2^(2^n-2) and describe the multiplication group Mlt(Q_n) as a semidirect product of Inn(Q_n)xZ_2 and an elementary abelian 2-group of order 2^n. We prove that one-sided inner mapping groups Inn_l(Q_n) and Inn_r(Q_n) are equal, elementa…
In an abelian group G, a more sums than differences (MSTD) set is a subset A of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets in finite abelian groups, extending previous results of Nathanson. The proof contains an application of a recently resolved conjecture of Alon and Kahn on the number of independent sets in a regular graph.
In this paper we prove that an abelian group contains $(2^{2m+1}(2^{m-1}+1), 2^m(2^m+1), 2^m)$-difference sets with $m\geqslant 3$ if and only if it contains an elementary abelian 2-group of order $2^{2m}$. Our proof shows that the method of constructing such difference sets is essentially unique.
Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.
We shortly present two small results regarding the study formal duality in finite abelian groups as introduced by Cohn, Kumar, Reiher and Schürmann. In particular, we give a new example of a formally self dual set in $\mathbb{Z}_2^2\times\mathbb{Z}_4^2$ and computed nonexistence of primitive formally dual sets of size $8$ in $\mathbb{Z}_8^2$.
Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)
We show that any connected Cayley graph $Γ$ on an Abelian group of order $2n$ and degree $\tildeΩ(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Γ$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.
We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.
We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.
