Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a(g) ∗ a(g) = 1H, because applying a to the first equality yields the second, and applying b to the second gives back the first. COUNT 3
QSN1 What are two groups called if they include homomorphisms? ANS1 isomorphic QSN2 What similar element do isomorphic groups carry? ANS2 isomorphic groups QSN3 How can showing the second equality yields the first prove the concept of isomorphic groups? ANS3 applying a to the first equality yields the second