We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group $G$ whose order satisfies certain arithmetic properties to the Cayley ...isomorphism problem of Cayley objects of the Sylow subgroups of $G$ in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that ${\mathbb Z}_q\times{\mathbb Z}_p^2\times{\mathbb Z}_m$ is a CI-group with respect to digraphs, where $q$ and $p$ are primes with $p^2 < q$ and $m$ is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on $q$ and $p$).
A finite group
R
is a
DCI
-group if, whenever
S
and
T
are subsets of
R
with the Cayley digraphs
Cay
(
R
,
S
)
and
Cay
(
R
,
T
)
isomorphic, there exists an automorphism
φ
of
R
with
S
φ
=
T
. The ...classification of
DCI
-groups is an open problem in the theory of Cayley digraphs and is closely related to the isomorphism problem for digraphs. This paper is a contribution toward this classification, as we show that every dihedral group of order 6
p
, with
p
≥
5
prime, is a
DCI
-group. This corrects and completes the proof of Li et al. (J Algebr Comb 26:161–181,
2007
, Theorem 1.1) as observed by the reviewer (Conder in Math review MR2335710).
A ternary relational structure
X
is an ordered pair
(
V
,
E
)
where
V
is a set and
E
a set of ordered 3-tuples whose coordinates are chosen from
V
(so a ternary relational structure is a natural ...generalization of a 3-uniform hypergraph). A ternary relational structure is called a Cayley ternary relational structure of a group
G
if
Aut
(
X
)
, the automorphism group of
X
, contains the left regular representation of
G
. We prove that two Cayley ternary relational structures of
Z
2
3
×
Z
p
,
p
≥
11
a prime, are isomorphic if and only if they are isomorphic by a group automorphism of
Z
2
3
×
Z
p
. This result then implies that any two Cayley digraphs of
Z
2
3
×
Z
p
are isomorphic if and only if they are isomorphic by a group automorphism of
Z
2
3
×
Z
p
,
p
≥
11
a prime.
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.
In this paper we are concerned with 3-groups. We prove that an elementary abelian 3-group of rank 5 is a
CI
(
2
)
-group, and that an elementary abelian 3-group of rank greater than or equal to 8 is ...not a
CI
-group. In Section 4, we present a conjecture.
The isomorphism problem for Cayley graphs has been extensively investigated over the past 30 years. Recently, substantial progress has been made on the study of this problem, many long-standing open ...problems have been solved, and many new research problems have arisen. The results obtained, and methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the finite simple group classification, through to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area.
A ternary relational structure X is an ordered pair (V,E) where V is a set and E a set of ordered 3-tuples whose coordinates are chosen from V (so a ternary relational structure is a natural ...generalization of a 3-uniform hypergraph). A ternary relational structure is called a Cayley ternary relational structure of a group G if inline image, the automorphism group of X, contains the left regular representation of G. We prove that two Cayley ternary relational structures of inline image, p>=11 a prime, are isomorphic if and only if they are isomorphic by a group automorphism of inline image. This result then implies that any two Cayley digraphs of inline image are isomorphic if and only if they are isomorphic by a group automorphism of inline image, p>=11 a prime.
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