Question

Let \( G \) be a group of order \( 5^4 \) with center having \( 5^2 \) elements. Then the number of conjugacy classes in \( G \) is \(\underline{\hspace{1cm}}\) .

Hint:

For groups of prime power order, use the class equation to determine the number of conjugacy classes. The size of each conjugacy class is a divisor of the order of the group.

Answer 1

Correct Answer: 145

Let the group \( G \) have order \( 5^4 = 625 \) and center \( Z(G) \) of order \( 5^2 = 25 \). The center of a group consists of elements that commute with all other elements, so all elements in \( Z(G) \) are in their own conjugacy classes. Thus, the number of conjugacy classes containing elements of the center is equal to the order of the center, which is 25. Now, by the class equation of finite groups, the order of the group is equal to the sum of the sizes of the conjugacy classes. The conjugacy classes not contained in the center must have more than one element. The size of each non-central conjugacy class is a divisor of the order of the group. We divide the remaining elements into conjugacy classes, and it turns out there are 120 conjugacy classes formed from these remaining elements. Thus, the total number of conjugacy classes is the sum of the 25 conjugacy classes from the center and the 120 conjugacy classes from the non-central elements: \[ \text{Total conjugacy classes} = 25 + 120 = 145. \] Thus, the number of conjugacy classes in \( G \) is \( \boxed{145} \).