Group Isomorphisms and Automorphisms

Prerequisite – Group

Isomorphism : 
For two groups (G,+) and (G’,*) a mapping f : G → G’ is called isomorphism if 

• f is one-one
• f is onto
• f is homomorphism i.e. f(a + b) = f(a) * f(b) ∀ a, b ∈ G.

In brief, a bijective homomorphism is an isomorphism.

Isomorphic group :
If there exists an isomorphism from group (G,+) to (G’,*). Then a group (G,+) is called isomorphic to a group (G’,*)
It is written as G ≅ G’ .

Examples –

1. f(x)=log(x) for groups (R⁺,*) and (R,+) is a group isomorphism.
Explanation – 

• f(x)=f(y) => log(x)=log(y) => x=y , so f is one-one.
• f(R⁺)=R , so f is onto.
• f(x*y)=log(x*y)=log(x)+log(y)=f(x)+f(y) , so f is a homomorphism.

 2. f(x)=ax for group (Z,+) to (aZ,+) , where a is any non zero no.
Explanation – 

• f(x)=f(y) => ax=ay => x=y , so f is one-one.
• f(Z)=aZ , so f is onto.
• f(x + y) =ax + ay= f(x) + f(y), so f is a homomorphism.  

 3. The function f from  group of cube roots of unity {} with a multiplication operation is an isomorphism to group residual classes mod(3) {{0},{1},{2}} with the operation of addition of residual classes mod(3) such that f(1)={0}, f()={1} and f()={2}.

Explanation –

• Clearly, f is onto and one-one.
• Also f(1*) = f() = {1} = {0} +₃ {1} = f(1)*f().
  f(*) = f(1) = {0} = {1} +₃ {2} = f()*f().
  and f(*1) = f() = {2}={2} +₃ {0} =  f()*f(1). So f is homomorphism.
  All this proves that f is an isomorphism for two referred groups.     

4. f(x)=e^x for groups (R,+) and (R+,*) where R+ is a group of positive real numbers and x is an integer.

5.Groups ({0,1,2,3},+₄) and ({2,3,4,1},+₅) are isomorphic.

NOTE :

1. If there is a Homomorphism f form groups (G,*) to (H,+) . Then f is also a Isomorphism if and only if Ker(f)={e} .Here e is the identity of (G,*). 
   Also, Ker(f) = Kernel of a homeomorphism f :(G,*) → (H,+) is a set of all the elements in G such that an image of all these elements in H is the identity element e’ of (H,+) .
2. If two groups are isomorphic, then both will be abelians or both will not be. Remember a group is Abelian if it is commutative.
3. A set of isomorphic group form an equivalence class and they have identical structure and said to be abstractly identical.

Automorphism : 
For a group (G,+), a mapping f : G → G is called automorphism if 

• f is one-one.
• f homomorphic i.e. f(a +b) = f(a) + f(b) ∀ a, b ∈ G.

Examples –

 1. For any group (G,+) an identity mapping I_g: G → G, such that I_g(g)=g , ∀g ∈ G is an automorphism.
Explanation-

• as if I(a)=I(b) => a=b so I is one-one.
• as I(a+b) =a+b =I(a)+I(b), so I is also a homomorphism.

2. f(x)=-x for group (Z,+).
Explanation- 

• as if f(a)=f(b) => -a=-b => a=b so f is one-one.
• as if f(a+b) =-(a+b) =(-a)+(-b) =f(a)+f(b), so f is also a homomorphism.                  

3. f(x)=axa^-1 for a group (G,+) ∀a ∈ G.
Explanation –

• as f(n)=f(m) => ana^-1 = ama^-1 => n = m so f is one-one.
• as f(n+m)= a(n+m)a^-1 =ana^-1 + ama^-1 = f(n) + f(m), so f is also  homomorphism.

4. f(z)= for groups of complex numbers with addition operation.
Remember f is complex conjugate such that if z=a+ib then f(z)===a-ib.

5.f(x)=1/x is automorphism for a group (G,*) if it is Abelian.

NOTE :

1. A set of all the automorphisms( functions ) of a group, with a composite of functions as binary operations forms a group.
2. Simply, an isomorphism is also called automorphism if both domain and range are equal.
3. If f is an automorphism of group (G,+), then (G,+) is an Abelian group.
4. Identity mapping as we see, in example, is an automorphism over a group is called trivial automorphism and other non-trivial.
5. Automorphism can be divided into inner and outer automorphism.