1.53: Group isomorphism
A group isomorphism is a special type of group homomorphism . It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic . Isomorphic groups have the same properties and the same structure of their multiplication table.
Let ( G , *) and ( H , #) be two groups, where "*" and "#" are the binary operations in G and H , respectively. A group isomorphism from ( G , *) to ( H , #) is a bijection from G to H , i . e . a bijective mapping f : G → H such that for all u and v in G one has
f ( u * v ) = f ( u ) # f ( v ).
Two groups ( G , *) and ( H , #) are isomorphic if an isomorphism between
them exists. This is written:( G , *) \(\cong\) ( H , #)
If H = G and the binary operations # and * coincide, the bijection is an automorphism .