Number of Conjugate Classes in a Group of Order 125

In order to determine the number of conjugate classes in a non-abelian group of order 125, we need to consider the structure of the group and apply relevant group theory concepts.

1. Group Structure:
A group of order 125 can be written as G = {e, a, a^2, ..., a^124} where e is the identity element and a is a non-identity element of the group. Since the group is non-abelian, the commutative property does not hold, i.e., ab ≠ ba for some elements a and b in the group.

2. Conjugate Elements:
Two elements a and b in a group G are said to be conjugate if there exists an element g in G such that b = g⁻¹ag. In other words, two elements are conjugate if they can be transformed into each other by a change of basis.

3. Conjugacy Class:
A conjugacy class is a set of all elements in a group that are conjugate to each other. It can be denoted as [a] = {g⁻¹ag | g ∈ G}, where a is an element of the group G.

4. Orbit-Stabilizer Theorem:
The orbit-stabilizer theorem states that the number of elements in a conjugacy class [a] is equal to the index of the centralizer of a in G, denoted as |[a]| = |G : C(a)|, where C(a) is the centralizer of a in G.

5. Centralizer:
The centralizer of an element a in a group G is the set of all elements that commute with a, i.e., C(a) = {g ∈ G | ga = ag}.

6. Determining the Number of Conjugate Classes:
For a non-abelian group, the number of conjugate classes is given by the number of distinct centralizers in the group.

In this case, we have a non-abelian group of order 125. Since the group is non-abelian, there exists at least one non-identity element a that does not commute with some other element b in G. This means that there are at least two distinct centralizers in the group, one for each conjugacy class.

Since the number of conjugate classes is equal to the number of distinct centralizers, the answer is option 'A' (29).