Question

# Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.

Abstract algebra
Let (Z,+) be a group of integers and (E,+) be a group of even integers. Find and prove if there exist an isomorphism between them.

2021-02-26

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.