2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77726Let G be a finite group. To every smooth G-action on a compact, connected and oriented surface we can associate its data of singular orbits. The set of such data becomes an Abelian group B_G under the G-equivariant connected sum. We will show that the map which sends G to B_G is functorial and carries many features of the representation theory of finite groups and thus describes a geometric representation theory. We will prove that B_G consists only of copies of Z and Z/2Z. Furthermore we will show that there is a surjection from the G-equivariant cobordism group of surface diffeomorphisms to B_G. We will define a G-signature which is related to the G-signature of Atiyah and Singer and prove that this new G-signature is injective on the copies of Z in B_G.28 pages, See also http://www.math.nwu.edu/~ralph/Algebraic TopologyGroup TheoryGeometric Topology57S25; 20C15; 57R85text