Let G be a group and S be a set. Then S is called a left G-set if there exists a map ϕ:G×S->S such that ϕ(g_1, ϕ(g_2, s)) = ϕ(g_1 g_2, s) for all s element S and all g_1, g_2 element G. This is commonly written ϕ(g, s) = g s, so the above relation becomes g_1(g_2 s) = (g_1 g_2) s. The map ϕ is called a left G-action on the set S. Right G-sets and right G-actions are defined analogously except elements of G are multiplied by elements of S to the right instead of to the left. Left G-sets and right G-sets are both called G-sets for simplicity.