Automorphism Groups of Cayley Maps

A *Cayley graph* has vertex set the elements of a group *G* and edge set determined by a balanced generating set *X*. A *Cayley map* *M* is an embedding of a Cayley graph in an oriented surface such that the local rotation at each vertex is described by the same permutation *p* of *X*.

**Question:** *Given a finite group* G, *is there a Cayley map* M *for which the* full *automorphism group* AutM *is isomorphic to* G?

Any finite group *G* ``lives inside'' *AutM* for any Cayley map *M=CM(G,X,p)* based on *G*. It is the adjective ``full'' that keeps the above question interesting. The Klein four-group, for example, is *not* the full automorphism group of any Cayley map.

The inner structure of automorphism groups of Cayley maps has been described in Theorem 4 of [J1] using the concept of a special graph-automorphism *rho*. Let *M=CM(G,X,p)*. The full automorphism group *AutM* of *M* is isomorphic to the group (*G* x <*rho*>, o), where the binary operation o is defined as (a, *rho*^m) o (b, *rho*^n) = (a rho^m(b), *theta*), and *theta* is a power of *rho* satisfying the property *theta*(c) = *rho*^m(b)^{-1} *rho*^m(b *rho*^n(c)), for all a,b,c in G.

Automorphism groups of Cayley maps are not semidirect products in general ( *rho* is not bound to be a group automorphism, in the first place), despite their obvious similarity. They are a special case of the more general *rotary product* described in [J2].

Based on this structural description, we obtain an immediate corollary toward the characterization of automorphism groups of Cayley maps. A *graphical regular representation* of group *G* is a Cayley graph *Gamma* =C(G,X) whose full automorphism group *Aut Gamma* is isomorphic to *G*. Any graph automorphism of *Gamma* which fixes a vertex is trivial. This simplifies the structure of the automorphism group of any Cayley map *CM(G,X,p)* based on *Gamma* and, in fact, makes it isomorphic to *G*. Thus, *any finite group* G *possessing a graphical regular representation* Gamma *is the full automorphism group of any Cayley map* M *based on* Gamma.

All the finite groups with graphical regular representation have been identified. A nice overview of this topic can be found in [G]. There are only two infinite classes and 13 sporadic cases of finite groups (the Klein group among them) that do not have a graphical regular representation. These are the cases whose structure has to be studied in order to complete the characterization of automorphism groups of Cayley maps.

References:

[G] C.D. Godsil, GRR's for non-solvable groups, Algebraic Methods in Graph Theory, vol.1, (L. Lovasz and T. Sos eds.), North-Holland 1981.

[J1] R.Jajcay, Automorphism Groups of Cayley Maps, *Journal of Comb. Theory Series B* **59** (1993) 297-310.

[J2] R.Jajcay, On a New Product of Groups, *Europ. J. Combinatorics* **15** (1994) 251-252.

Submitted by: Robert Jajcay

Send comments to *dan.archdeacon@uvm.edu *and *jajcay@helios.unl.edu*

August, 1995