Two topological spaces that are homotopy equivalent have isomorphic homology groups. It provides a general definition of isomorphism, and justifies transfer of properties between isomorphic structures. Each is a matroid invariant, which means that isomorphic matroids have the same polynomial. The Oberwolfach problem concerns the existence of 2-factorizations of complete graphs into isomorphic subgraphs. The fact that the polynomial representation is unique implies that is isomorphic to . Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H. Every infinite cyclic group is isomorphic to Z. For example, and are isomorphic (but "not" isomorphic as topological fields). A "relation-number" is an equivalence class of isomorphic relations. Thus is isomorphic to the unit quaternions. The tensor product can be shown to be isomorphic to . Then is isomorphic to the semidirect product . It follows that has Galois group isomorphic to over . This approach reduces the search space by formula 88 by eliminating isomorphic automata. These coordinate vectors form another vector space, which is isomorphic to the original vector space.