I want to show that $$U(3)×U(5)\cong U(15)$$ Would I simply have to find an isomorphic map that maps the two groups, or is there a clever way to approach this? I have been trying to find an isomorphic map but have had no luck. Does anyone know of an isomorphic map that maps these two groups?
$U(3) \times U(5) = \left \{ (1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)\right \} $ and $U(15) = \left \{ 1,2,4,7,8,11,13,14\right \}$