Introduction:
In this question, we are given a group that has 5 elements of order 2. We need to determine whether this group can be abelian or not.

Understanding the problem:
To solve this problem, we need to understand the concept of order and abelian groups.

Order of an element:
The order of an element in a group is the smallest positive integer n such that the element raised to the power of n gives the identity element of the group. In other words, the order of an element is the number of times we need to apply the group operation to get the identity element.

Abelian groups:
An abelian group, also known as a commutative group, is a group in which the group operation is commutative. In other words, for any two elements a and b in the group, the result of the operation ab is the same as the result of the operation ba.

Analysis:
Let's assume that our group G has 5 elements of order 2, denoted by a, b, c, d, and e. Since the order of each element is 2, we have a^2 = b^2 = c^2 = d^2 = e^2 = e (identity element).

If G is an abelian group, then for any two elements a and b in G, ab = ba. Let's consider two arbitrary elements a and b in G.

Proof:
If a and b are distinct elements, then ab and ba must be distinct as well. This is because if ab = ba, then multiplying both sides by a^-1 on the left gives b = a^-1ba = a^-1ab = b, which contradicts the assumption that a and b are distinct.

Now, let's consider the product ab. Since a and b are distinct elements, ab cannot be equal to a or b because the order of a and b is 2. Similarly, the product ba cannot be equal to a or b. Therefore, ab and ba must be distinct elements in the group.

However, we have assumed that our group has 5 elements of order 2. This means that there are only 5 distinct elements in the group. Since ab and ba are distinct, this implies that our group must have at least 6 distinct elements, which contradicts the given information.

Conclusion:
Therefore, it is not possible for a group with 5 elements of order 2 to be abelian.