We generalize the idea of reflexibility of a map on a surface by introducingcertain integers as its ”exponents“. An exponent is any integer $e$ withthe property that changing the cyclic permutation of edges around eachvertex induced by the map to its $e$-th power gives rise to an isomorphicmap. The exponents reduced modulo the least common multiple of the vertexvalencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of themap $M$. Along with the automorphism group, the group in fact provides anadditional measure of symmetry of ~$M$.

The paper is devoted to developing the fundamentals of the theory ofexponent groups of maps. Motivation comes from the problem of classificationof regular maps with a given underlying graph. To this end, we prove thatthe number of non-isomorphic regular maps (if any) with a given underlyinggraph and the same map automorphism group is $|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. Wecalculate the exponent groups for some interesting families of regular mapsincluding complete maps. Special attention is paid to the problem of how theantipodality of a map is reflected by its exponent group. In the finalsection we discuss several open problems.

1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.