Explanation:

The given conditions in the group G are:
1. ab = a
2. ba = a

We need to determine the correct statement among the given options.

Let's consider the first condition, ab = a. Multiplying both sides of this equation by the inverse of 'b' (denoted as b⁻¹), we get:

ab * b⁻¹ = a * b⁻¹

This simplifies to:

a * (b * b⁻¹) = a * b⁻¹

Since b * b⁻¹ is the identity element (denoted as e) in the group G, the equation becomes:

a * e = a * b⁻¹

And since a * e = a, we have:

a = a * b⁻¹

This result shows that the inverse of 'b' in the group G is equal to 'a'. Hence, option 'a' is incorrect.

Now, let's consider the second condition, ba = a. Multiplying both sides of this equation by the inverse of 'a' (denoted as a⁻¹), we get:

b * a * a⁻¹ = a * a⁻¹

This simplifies to:

b * (a * a⁻¹) = a * a⁻¹

Since a * a⁻¹ is the identity element (denoted as e) in the group G, the equation becomes:

b * e = a * a⁻¹

And since b * e = b, we have:

b = a * a⁻¹

This result shows that the inverse of 'a' in the group G is equal to 'b'. Hence, option 'b' is incorrect.

Now, let's consider the third condition, b = ed. Multiplying both sides of this equation by the inverse of 'd' (denoted as d⁻¹), we get:

b * d⁻¹ = ed * d⁻¹

This simplifies to:

b * (d * d⁻¹) = e * d⁻¹

Since d * d⁻¹ is the identity element (denoted as e) in the group G, the equation becomes:

b * e = e * d⁻¹

And since b * e = b, we have:

b = e * d⁻¹

This result shows that the inverse of 'd' in the group G is equal to 'b'. Hence, option 'd' is incorrect.

Therefore, the correct statement among the given options is option 'c': a² = e.