Isomorphism between G and H under addition:

Introduction:
In order to determine whether the groups G and H are isomorphic under addition, we need to examine their structural properties and find a suitable mapping between them that preserves the group operation.

Group G:
The group G is defined as G = {a + b√2 | a, b ∈ rational numbers}. It consists of all rational numbers of the form a + b√2, where a and b are rational numbers.

Group H:
The group H is defined as H = {[a 2b b a] | a, b ∈ rational numbers}. It consists of all 2x2 matrices of the form [a 2b b a], where a and b are rational numbers.

Mapping between G and H:
To establish an isomorphism between G and H, we need to find a mapping that preserves the group operation (addition) between the elements of G and H. Let's consider the mapping φ: G → H.

Mapping the identity:
The identity element in G is 0, as adding 0 to any element in G does not change its value. The identity element in H is the matrix [0 0 0 0], which is also the additive identity for matrices. Therefore, we can map the identity element of G to the identity element of H, i.e., φ(0) = [0 0 0 0].

Mapping addition:
For the mapping to preserve the group operation, it must satisfy φ(a + b√2) = φ(a) + φ(b√2) for all a, b ∈ rational numbers. Let's consider two arbitrary elements x, y ∈ G, where x = a + b√2 and y = c + d√2.

We can express x and y in matrix form as:
x = [a b√2]
y = [c d√2]

To find the mapping φ(x) and φ(y), we need to determine the corresponding matrices in H. By observing the structure of the matrices in H, we can map x and y as follows:
φ(x) = [a 2b b a]
φ(y) = [c 2d d c]

To show that φ preserves addition, we need to demonstrate that φ(x + y) = φ(x) + φ(y). Let's compute the addition on both sides:

x + y = [a + b√2 + c + d√2]
= [a + c + (b + d)√2]

φ(x + y) = [a + c 2(b + d) (b + d) a + c]
= [a 2b b a] + [c 2d d c]
= φ(x) + φ(y)

Conclusion:
We have shown that there exists a mapping φ between G and H that preserves addition, i.e., φ(x + y) = φ(x) + φ(y) for all x, y ∈ G. Therefore, G and H are isomorphic under addition.