Order of a group

The order of a group refers to the number of elements in the group. In this case, we are given that the order of the group G is 125.

Center of a group

The center of a group G, denoted as Z(G), is the set of elements that commute with every element of the group. In other words, for any element g in G, if h is in the center of G, then gh = hg.

Calculating the order of the center of G

To calculate the order of the center of G, we need to find the number of elements in Z(G).

Property 1: The identity element of the group G is always in the center of G.

Since the identity element commutes with every element of the group, it must be in the center. Therefore, Z(G) contains at least one element.

Property 2: If G is a group of prime order, then the center of G is the entire group.

In our case, the order of G is 125, which is not a prime number. Therefore, we cannot conclude that the center of G is the entire group.

Property 3: If G is a group of order p^n, where p is a prime number and n is a positive integer, then the center of G has at least p elements.

Since the order of G is 125, we can write it as 5^3. According to Property 3, the center of G must have at least 5 elements.

Property 4: The order of the center of G must divide the order of G.

In this case, the order of the center of G must divide 125. The only factor of 125 that is less than or equal to 5 is 5 itself. Therefore, the order of the center of G is 5.

Conclusion

The order of the center of G is 5, based on the properties of group theory.