Proof:

1. Introduction:
We are given that G is an abelian group with an odd number of elements. We need to prove that the product of all the elements in G is the identity element of G.

2. Properties of Abelian Group:
Let's recall some properties of an abelian group:
- Closure: For any two elements a and b in G, the product a * b is also in G.
- Associativity: For any three elements a, b, and c in G, (a * b) * c = a * (b * c).
- Identity Element: There exists an element e in G such that for any element a in G, a * e = e * a = a.
- Inverse Element: For every element a in G, there exists an element b in G such that a * b = b * a = e.

3. Product of All Elements:
Let's denote the product of all the elements in G as P. Since G is an abelian group, we can rearrange the elements in any order without changing the product.

4. Pairing of Elements:
We can pair each element with its inverse to get the identity element. Since G has an odd number of elements, there will be one element left without a pair.

5. Product of Pairs:
Let's consider the product of each pair of elements. Each pair consists of an element a and its inverse b. The product of the pair is a * b = e, the identity element.

6. Remaining Element:
We have one element left without a pair, let's denote it as x. Since G is an abelian group, x * x^(-1) = e, where x^(-1) is the inverse of x. Therefore, x^2 = e.

7. Product of All Elements:
Now, let's consider the product P = (a1 * b1) * (a2 * b2) * ... * (xn-1 * bn-1) * x, where a1, a2, ..., xn-1 are the elements in pairs and b1, b2, ..., bn-1 are their inverses.

8. Associativity:
By the associativity property of the abelian group, we can rearrange the product as P = (a1 * a2 * ... * xn-1 * x) * (b1 * b2 * ... * bn-1).

9. Identity Element:
Since the product of each pair is the identity element, the product of all the elements in the parentheses is e.

10. Conclusion:
Therefore, the product of all the elements in G is the identity element e, which proves the given statement.