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.
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 = 1 G 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 .
How can showing the second equality yields the first prove the concept of isomorphic groups?
How can showing the second equality yields the first prove the concept of isomorphic groups ?
applying a to the first equality yields the second
applying a to the first equality yields the second