Summary:
The minimum number of elements of the group G, which are their own inverse, is 1.

Explanation:
To find the minimum number of elements in G that are their own inverses, we need to understand what it means for an element to be its own inverse.

Definition of an Inverse:
In a group, an inverse of an element x is another element y such that their product x * y equals the identity element (usually denoted as e). In other words, if x * y = e, then y is the inverse of x.

Identity Element:
The identity element is an element in the group such that when it is multiplied by any other element, it doesn't change the value of that element. In other words, for any element x in the group, x * e = e * x = x.

Number of Elements in G:
Given that the group G has 10 elements, let's assume that there are n elements in G that are their own inverses.

Case 1: n = 0
If there are no elements in G that are their own inverses, it means that for every element x in G, there exists another element y that is the inverse of x. Therefore, for every element x, x * y = e, where y is the inverse of x. This implies that y is unique for each x in G.

Since G has 10 elements and each element has a unique inverse, it means that there are at least 10 different elements in G. However, this contradicts the given information that G has only 10 elements. Hence, this case is not possible.

Case 2: n ≥ 1
If there is at least one element in G that is its own inverse, let's call it x. This means that x * x = e, where e is the identity element. Since x is its own inverse, there are no other elements in G that can be the inverse of x.

Now, for every other element y in G, y * x cannot be equal to e, because if it were, then y would also be the inverse of x, which is not possible.

Therefore, for n elements in G that are their own inverses (where n ≥ 1), we need at least n unique elements other than x in G. Since G has 10 elements, it means that the minimum value of n is 1.

Conclusion:
The minimum number of elements in G that are their own inverses is 1.