In this paper, we prove that the group
Z
p
n
is not a CI-group if
n⩾2p−1+(
2p−1
p
)
, that is there exist two Cayley digraphs over
Z
p
n
which are isomorphic but their connection sets are not ...conjugate by an automorphism of
Z
p
n
.
The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete ...classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.
A ternary relational structure
X is an ordered pair (
V,
E), where
E⊂
V
3. A ternary relational structure
X is a Cayley ternary relational structure of a group
G if the left regular representation of
...G is contained in the automorphism group of
X. A group
G is a CI-group with respect to ternary relational structures if whenever
X and
X′ are isomorphic Cayley ternary relational structures of
G, then
X and
X′ are isomorphic if and only if they are isomorphic by an automorphism of
G. In this paper, we will provide a (relatively short) list of all possible CI-groups with respect to color ternary relational structures. All of these groups have order 2
d
n, 0⩽
d⩽5, and
n a positive integer with gcd(
n,
ϕ(
n))=1, where
ϕ is Euler's phi function. If
d=0, it has been shown by Pálfy that
Z
n
is a CI-group with respect to every class of combinatorial objects. We then show that of the possible CI-groups with respect to color ternary relational structures of order 2
n and 4
n with a cyclic Sylow 2-subgroup, most are CI-groups with respect to ternary relational structures, and in the unresolved cases, give a necessary and sufficient condition for the group to be a CI-group with respect to ternary relational structures. In particular, we determine all cyclic CI-groups with respect to ternary relational structures. Finally, a group
G that is a CI-group with respect to ternary relational structures is also a CI-group with respect to binary relational structures (i.e. graphs and digraphs). Some of the groups considered in this paper are not known to be CI-groups with respect to graphs or digraphs, and we thus provide new examples of CI-groups with respect to graphs and digraphs.
LetG be a finite group, andS a subset ofG \ |1| withS =S−1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) ...there is an α∈ Aut(G) such thatSα =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S−1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA5 is a 4-Cl-group, which was a conjecture posed by Li and Praeger.
The Group is a CI-Group Kovács, I.; Muzychuk, M.
Communications in algebra,
10/9/2009, Volume:
37, Issue:
10
Journal Article
Peer reviewed
In this article it is proven that the group
is a CI-group, that is two Cayley graphs over
are isomorphic if and only if their connection sets are conjugate by an automorphism of the group
.
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