We generalize the idea of reflexibility of a map on a surface by introducing
certain integers as its ”exponents“. An exponent is any integer $e$ with
the property that changing the cyclic permutation of edges around each
vertex induced by the map to its $e$-th power gives rise to an isomorphic
map. The exponents reduced modulo the least common multiple of the vertex
valencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of the
map $M$. Along with the automorphism group, the group in fact provides an
additional measure of symmetry of ~$M$.
The paper is devoted to developing the fundamentals of the theory of
exponent groups of maps. Motivation comes from the problem of classification
of regular maps with a given underlying graph. To this end, we prove that
the number of non-isomorphic regular maps (if any) with a given underlying
graph and the same map automorphism group is
$|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. We
calculate the exponent groups for some interesting families of regular maps
including complete maps. Special attention is paid to the problem of how the
antipodality of a map is reflected by its exponent group. In the final
section we discuss several open problems.
1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.