# Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants

@article{Li2005PermutationGW, title={Permutation Groups with a Cyclic Regular Subgroup and Arc Transitive Circulants}, author={Caiheng Li}, journal={Journal of Algebraic Combinatorics}, year={2005}, volume={21}, pages={131-136} }

A description is given of finite permutation groups containing a cyclic regular subgroup. It is then applied to derive a classification of arc transitive circulants, completing the work dating from 1970’s. It is shown that a connected arc transitive circulant Γ of order n is one of the following: a complete graph Kn, a lexicographic product $\Sigma [{\bar K}_b]$, a deleted lexicographic product $\Sigma [{\bar K}_b] - b\Sigma$, where Σ is a smaller arc transitive circulant, or Γ is a normal…

## 80 Citations

### A note on arc-transitive circulant digraphs

- Mathematics
- 2008

Abstract We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting…

### On Prime-Valent Symmetric Bicirculants and Cayley Snarks

- MathematicsGSI
- 2013

It is shown that a connected bicirculant X ≠ K 4 of prime valency admitting a group of automorphisms containing a (2,n)-semiregular automorphism and acting regularly on the set of arcs is near-bipartite (that is, with the chromatic number at most 3).

### Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2019

Abstract A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on…

### On Certain Edge-Transitive Bicirculants

- MathematicsElectron. J. Comb.
- 2019

It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc- transitive as well as infinitely many half-arc-transitives members are identified.

### Arc-transitive Dihedrants of Odd Prime-power Order

- MathematicsGraphs Comb.
- 2013

The connected, arc-transitive Cayley graphs are classified as the dihedral group of order 2pn, p is an odd prime.

### On the automorphism groups of almost all circulant graphs and digraphs

- MathematicsArs Math. Contemp.
- 2014

The first author has conjectured that almost all circulant graphs have automorphism groups as small as possible, but it is shown that this conjecture is not true in general, but is true if the authors consider only those circulants (di)graphs whose order is in a "large" subset of integers.

### Tetravalent arc-transitive locally-Klein graphs with long consistent cycles

- MathematicsEur. J. Comb.
- 2014

### Digraph Representations of 2-closed Permutation Groups with a Normal Regular Cyclic Subgroup

- MathematicsElectron. J. Comb.
- 2015

2-closed permutation groups which contain a normal regular cyclic subgroup and prove that for each such group $G$ there exists a circulant $\Gamma$ such that $\mathrm{Aut} (\Gamma)=G$.

## References

SHOWING 1-10 OF 19 REFERENCES

### Classifying Arc-Transitive Circulants of Square-Free Order

- Mathematics
- 2001

AbstractA circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classified. It is shown that an arc-transitive circulant Γ of square-free order n is one of…

### Finite edge-transitive Cayley graphs and rotary Cayley maps

- Mathematics
- 2006

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal,…

### A Classification of 2-Arc-Transitive Circulants

- Mathematics
- 1996

A graph X is k-arc-transitive if its automorphism group acts transitively on the set of k-arcs of X. A circulant is a Cayley graph of a cyclic group. A classification of 2-arc-transitive circulants…

### Classifying Arc-Transitive Circulants

- Mathematics
- 2004

A circulant is a Cayley digraph over a finite cyclic group. The classification of arc-transitive circulants is shown. The result follows from earlier descriptions of Schur rings over cyclic groups.

### On the full automorphism group of a graph

- MathematicsComb.
- 1981

It is proved here that for the certain transitive permutation groups a simple necessary condition is also sufficient, and that, whenG is ap-group with no homomorphism ontoZp wrZp, almost all Cayley graphs ofG have automorphism group isomorphic toG.

### The Finite Primitive Permutation Groups Containing an Abelian Regular Subgroup

- Mathematics
- 2003

A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long‐standing open problem in permutation group theory initiated by…

### On Schur Rings over Cyclic Groups, II

- Mathematics
- 1996

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σg eHg Σg∈Hg ∉S, thenS is…

### On schur rings over cyclic groups

- Mathematics
- 1998

In this paper, we introduce the notion of wedge product of Schur rings. We show that for any nontrivial Schur ringS over a cyclic groupG, if there is a subgroupH such that Σg εHg Σg∈Hg ∉S, thenS is…