To prove any two groups are isomorphic:

The map \(\displaystyle\phi:Г\rightarrowΓ′\) is called an isomorphism \(Γ\) and \(Γ'\) and are

said to be isomorphic if

i) \(\displaystyle\phi\) is a homomorphism.

ii) \(\displaystyle\phi\) is bijective.

\(\displaystyleΓ\stackrel{\sim}{=}Γ′\) denotes \(Γ\) is isomorphic to \(Γ′\)

Let \(\displaystyle\phi:{Z}\rightarrow{2}{Z}\) be defined by \(\displaystyle{x}\mapsto{x}+{x}={2}{x}\)

i) To prove \(\phi\) is homomorphism:

\(\displaystyle\phi{\left({x}+{y}\right)}=\phi{\left({x}\right)}+\phi{\left({y}\right)}\)

\(\displaystyle\phi{\left({x}+{y}\right)}={2}{\left({x}+{y}\right)}\)

\(\displaystyle\phi{\left({x}+{y}\right)}={2}{\left({x}\right)}+{2}{\left({y}\right)}\)

\(\displaystyle\phi{\left({x}+{y}\right)}=\phi{\left({x}\right)}+\phi{\left({y}\right)}\)

Therefore, \(\displaystyle\phi\) is homomorphism

ii) To prove \(\displaystyle\phi\) is bijective:

Bijective is one-one and onto

To prove phi is one -one:

"Let us consider ‘f’ is a function whose domain is set A. The function is said to be injective(1-1) if for all x and y in A,

\(f(x)=f(y)\), then \(x=y''\)

Let,

\(\displaystyle\phi{\left({x}\right)}=\phi{\left({y}\right)}\)

\(\displaystyle{2}{x}={2}{y}\)

\(\displaystyle\Rightarrow{x}={y}\)

Therefore, \(\displaystyle\phi\) is one-one.

To prove \(\displaystyle\phi\) is onto:

"A function f from a set X to a set Y is surjective (also known as onto, or a surjection), if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that \(f(x)=y''\).

Let \(\displaystyle{y}\in{2}{Z}\)

Then y=2k for some \(\displaystyle{k}\in{Z}\)

Since \(\displaystyle{k}\in{Z}\) and

\(\displaystyle\phi{\left({k}\right)}={2}{k}\)

\(\displaystyle\Rightarrow{y}\)

\(\displaystyle\phi\) is onto

\(\displaystyle\Rightarrow\phi\) is bijective.

Since, it is homomorphism and bijective.

it is isomorphic.

\(\displaystyleΓ\stackrel{\sim}{=}Γ′\)

Therefore,The group (Z,+) and (E,+) are isomorphic.