# 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 i*s*onto**f i*s*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 resedual classes mod(3) {{0},{1},{2}} with the operation of addition of resudual 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 :**

- 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,+) . - If two groups are isomorphic, then both will be abelians or both will not be. Remember a group is Abelian if it is commutative.
- 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.**Explaination-**

- 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,+).**Explaination- **

- 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.**Explaination –**

- 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 homomorhism.

4. f(z)= for groups of complex numbers with addition operation.

Remember f is complex comjugate 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 :**

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