Group Isomorphisms and Automorphisms

In the study of algebraic structures, group isomorphisms and automorphisms play a fundamental role. By defining internal symmetries inside a group (automorphisms) and when two groups have the same structure (isomorphisms), these ideas aid in our understanding of the structure and symmetry of groups.

Definition of Group Isomorphism

For two groups (G,+) and (G',*) a mapping f: G → G' is called an 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'.

Properties of Isomorphisms

• Bijectiveness: An isomorphism is both injective (one-to-one) and surjective (onto), which makes it a bijection.

• Preservation of Structure: Group operations are preserved by isomorphisms, which means that the target group's operation is the image of the original group's operation under the isomorphism.

Examples of Isomorphisms

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.w
• 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 {1,w,w^2     } 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(w     )={1} and f(w^2)={2}.

Explanation

• Clearly, f is onto and one-one.
• Also f(1*w     ) = f(w     ) = {1} = {0} +₃ {1} = f(1)*f(w     ).
  f(w     *w^2     ) = f(1) = {0} = {1} +₃ {2} = f(w     )*f(w^2     ).
  and f(w^2     *1) = f(w^2     ) = {2}={2} +₃ {0} =  f(w^2     )*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

Definition of Group 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.

Properties of Automorphisms

• Identity Automorphism: The identity mapping Ig: G → G, defined by Ig(g)=g ,g∀g ∈ G is an automorphism.

• Inverse Automorphisms: Every automorphism has an inverse which is also an automorphism.

• Composition: The composition of two automorphisms is also an automorphism.

Examples of Automorphisms

 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)={\displaystyle {\overline {z}}}      for groups of complex numbers with addition operation.
Remember f is complex conjugate such that if z=a+ib then f(z)={\displaystyle {\overline {z}}}     ={\displaystyle {\overline {a+ib}}}     =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.

Relationship Between Isomorphisms and Automorphisms

• Automorphism as Isomorphism: An automorphism is an isomorphism that exists between a group and itself.

• Group Structure Preservation: Automorphisms particularly translate a group to itself, although isomorphisms also maintain the group structure.

• Equivalence Classes: An equivalence class is a collection of isomorphic groups that are considered to be abstractly similar since they have the same structures.

Group Isomorphisms and Automorphisms - Solved Examples

Example - 1 : Prove that (Z,+) and (2Z,+) are isomorphic.

Solution:

• Define f:Z→2Z by f(x)=2x.
• One-one: f(x)=f(y)⇒2x=2y⇒x=y.
• Onto: For every y∈2Z, there exists x∈Z such that y=2x.
• Homomorphism: f(x+y)=2(x+y)=2x+2y=f(x)+f(y).
• Hence, f is an isomorphism.

Example - 2: Show that (R⁺ ,⋅) and (R,+) are isomorphic via f(x)=log(x).

Solution:

• One-one: f(x)=f(y)⇒log(x)=log(y)⇒x=y.
• Onto: For every y∈R, there exists x∈R⁺ such that y=log(x).
• Homomorphism: f(x⋅y)=log(x⋅y)=log(x)+log(y)=f(x)+f(y).
• Hence, f is an isomorphism.

Example -3: Verify that the groups (Z₆,+₆) and (Z₂×Z₃,+) are isomorphic.

Solution:

• Define f:Z₆→Z₂×Z₃ by f(x)=(xmod2,xmod3).
• One-one: Suppose f(x)=f(y). Then (xmod2,xmod3)=(ymod2,ymod3).This implies x≡ymod6, hence x=y.
• Onto: For every 𝑍₃(a,b)∈Z₂×Z₃, there exists x∈Z₆ such that xmod2=a and xmod3=b.
• Homomorphism: f(x+₆​y)=((x+y)mod2,(x+y)mod3)=f(x)+f(y).

Hence, f is an isomorphism.

Example -4: Determine if the mapping f(x)=2x from (R,+) to (R,+) is an automorphism.

Solution:

• Define f:R→R by f(x)=2x.
• One-one: Suppose f(x)=f(y). Then 2x=2y. This implies x=y.
• Onto: For every y∈R, there is no x∈R such that 2x=y (specifically, if y is odd, there is no integer x such that 2x=y).
• Homomorphism: f(x+y)=2(x+y)=2x+2y=f(x)+f(y).

Since f is not onto, it is not an automorphism.

Example - 5: Check if the function f(x)=x²defines an isomorphism between the groups (R,+) and (R⁺,⋅).

Solution:

• Define f:R→R⁺ by f(x)=x².
• One-one: Suppose f(x)=f(y).Then x²=y².This implies x=±y, so f is not one-one.
• Onto: For every y∈R⁺, choose x= √y ​or x=− √yin R.
• Homomorphism: f(x+y)=(x+y)²=x²+2xy+y²≠f(x)⋅f(y).

Since f is neither one-one nor a homomorphism, it is not an isomorphism.

Practice Problems - Group Isomorphisms and Automorphisms

1. Prove that the groups (Z₄,+₄) and (Z₂×Z₂,+) are isomorphic.

2. Show that (R⁺,⋅) and (R,+) are isomorphic via f(x)=e^x.

3. Verify that the groups (Z₆,+₆) and (Z₂×Z₃,+) are isomorphic.

4. Determine if the mapping f(x)=2x from (R,+) to (R,+) is an automorphism.

5. Check if the function f(x) = x² defines an isomorphism between the groups (R,+) and (R⁺,⋅).

6. Show that (Z,+) and (3Z,+) are isomorphic.

7. Prove that the function f(x)=−x is an automorphism of the group (Z,+).

8. Verify that the function f(z) = \bar{z} is an automorphism for the group of complex numbers under addition.

9. Determine if the function f(x)=x⁻¹ is an automorphism for the group (R^∗,⋅).

10. Show that (Z_n,+_n) and (Z,+) are isomorphic when n is a prime number.