17 research outputs found
Abelian Carter subgroups in finite permutation groups
We show that a finite permutation group containing a regular abelian
self-normalizing subgroup is soluble.Comment: 6 page
Distinguishing Chromatic Number of Random Cayley graphs
The \textit{Distinguishing Chromatic Number} of a graph , denoted
, was first defined in \cite{collins} as the minimum number of
colors needed to properly color such that no non-trivial automorphism
of the graph fixes each color class of . In this paper, we
consider random Cayley graphs defined over certain abelian groups
and show that with probability at least we have,
.Comment: 11 page
Automorphisms of Cayley graphs on generalised dicyclic groups
A graph is called a GRR if its automorphism group acts regularly on its
vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that
there are only two infinite families of finite groups that do not admit GRRs :
abelian groups and generalised dicyclic groups. Indeed, any Cayley graph on
such a group admits specific additional graph automorphisms that depend only on
the group. Recently, Dobson and the last two authors showed that almost all
Cayley graphs on abelian groups admit no automorphisms other than these obvious
necessary ones. In this paper, we prove the analogous result for Cayley graphs
on the remaining family of exceptional groups: generalised dicyclic groups.Comment: 18 page
Full automorphism group of commuting and non-commuting graph of dihedral and symmetric groups
An automorphism of finite graph G is a permutation on its vertex set that conserves adjacency. The set of all automorphism of G is a group under composition of function. This group is called the full automorphism group of G. Study on the full automorphism group is an interesting topic because most graphs have only the trivial automorphism and many special graphs have many automorphisms. One of the special graphs is graph that associated with group. The result of this paper is the full automorphism groups of commuting and non-commuting graph of non-abelian finite group, especially on dihedral and symmetric groups, according to the choice of their subgroups
Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power order
We show that almost every Cayley graph ▫▫ of an abelian group ▫▫ of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph ▫▫ of an abelian group ▫▫ of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of ▫▫ (that is, ▫▫