Rédei and Megyesi proved that the number of directions determined by a
p
-element subset of
F
p
2
is either 1 or at least
p
+
3
2
. The same result was independently obtained by Dress, Klin, and ...Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.
A well-known conjecture of Babai states that if
G
is any finite simple group and
X
is a generating set for
G
, then the diameter of the Cayley graph Cay(
G
,
X
) is bounded by log ∣
G
∣
c
for some ...universal constant
c
. In this paper, we prove such a bound for Cay(
G
,
X
) for
G
= PSL(
n
,
q
), PSp(
n
,
q
) or PSU(
n
,
q
) where
q
is odd, under the assumptions that
X
contains a transvection and
q
≠ 9 or 81.
The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Łaba and Meyerowitz, but the spectral-tile direction is far from ...being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede’s conjecture holds in this case.
An
n
-ary associative function is called reducible if it can be written as a composition of a binary associative function. We summarize known results when the function is defined on a chain and is ...nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. ...This proves that these graphs do not form a family of expanders.
We investigate monotone idempotent
n
-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral ...element. We also present a complete characterization of idempotent monotone
n
-associative functions on an interval that have neutral elements.
For every prime
p
>2 we exhibit a Cayley graph on
which is not a CI-graph. This proves that an elementary abelian
p
-group of rank greater than or equal to 2
p
+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 of Muzychuk and Spiga concerning the problem.
The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored ...graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex.Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group $H$ is a CIM-group if any two Cayley maps over $H$ are isomorphic if and only if they are Cayley isomorphic.The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following$$\mathbb{Z}_m\times\mathbb{Z}_2^r, \\mathbb{Z}_m\times\mathbb{Z}_{4},\\mathbb{Z}_m\times\mathbb{Z}_{8}, \ \mathbb{Z}_m\times Q_8, \\mathbb{Z}_m\rtimes\mathbb{Z}_{2^e}, e=1,2,3,$$ where $m$ is an odd square-free number and $r$ a non-negative integer. Our second main result shows that the groups $\mathbb{Z}_m\times\mathbb{Z}_2^r$, $\mathbb{Z}_m\times\mathbb{Z}_{4}$, $\mathbb{Z}_m\times Q_8$ contained in the above list are indeed CIM-groups.
PDF
