Isomorphism between the group of integers under addition and the group of even integers under addition:

Definition of Isomorphism:
An isomorphism is a bijective map between two groups that preserves the group operation. It means that if G and H are two groups, then an isomorphism between G and H is a function f: G -> H such that:
1. f is a bijection (one-to-one and onto).
2. For all a, b in G, f(a * b) = f(a) + f(b).

Group of Integers under Addition:
The group of integers under addition, denoted by (Z, +), consists of all the integers and the operation is addition. The group operation of addition is associative, has an identity element 0, and every element has an inverse. It is an example of an abelian group.

Group of Even Integers under Addition:
The group of even integers under addition, denoted by (2Z, +), consists of all the even integers and the operation is addition. The group operation of addition is associative, has an identity element 0, and every element has an inverse. It is also an example of an abelian group.

Constructing the Isomorphism:
To find an isomorphism between (Z, +) and (2Z, +), we need to define a function f: Z -> 2Z that satisfies the two conditions mentioned earlier.

Let's define the function f as follows:
f(n) = 2n, for all n in Z.

Showing f is a Bijective Map:
To show that f is a bijection, we need to prove that it is both one-to-one and onto.

1. One-to-One (Injective):
Assume f(a) = f(b), where a, b are integers. Then, 2a = 2b, which implies a = b. Therefore, f is injective.

2. Onto (Surjective):
For any even integer y in 2Z, let's consider n = y/2. Then, f(n) = 2n = 2(y/2) = y. Therefore, f is surjective.

Since f is both injective and surjective, it is a bijection.

Preservation of Group Operation:
Next, we need to show that f preserves the group operation. That is, for any a, b in Z, f(a + b) = f(a) + f(b).

Let's consider a, b in Z. Then,
f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b).

Therefore, f preserves the group operation.

Conclusion:
The function f: Z -> 2Z defined as f(n) = 2n is an isomorphism between the group of integers under addition (Z, +) and the group of even integers under addition (2Z, +). It is a bijection and preserves the group operation, satisfying the conditions required for an isomorphism.