Final answer:
The relation of being conjugate is an equivalence relation on G. The conjugacy classes that consist of only one element are the trivial conjugacy classes, which contain the elements that commute with every element in G.
Step-by-step explanation:
The relation of being conjugate is an equivalence relation on G if it satisfies three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity: For any element a in G, a is conjugate to itself. This is because every element is conjugate to itself by the identity element. So, the relation is reflexive.
2. Symmetry: If a is conjugate to b, then b is conjugate to a. This is because if a = gbg^(-1), then b = g^(-1)ag. So, the relation is symmetric.
3. Transitivity: If a is conjugate to b and b is conjugate to c, then a is conjugate to c. This is because if a = gbg^(-1) and b = hch^(-1), then a = ghch^(-1)g^(-1), which shows that a is conjugate to c. So, the relation is transitive.
Conjugacy classes consist of elements that are conjugate to each other. The conjugacy class that consists of only one element is called the trivial conjugacy class. It contains the elements that commute with every element in G. In other words, if an element a in G is in a conjugacy class of size one, then a commutes with every element in G.
For example, in the group of integers under addition, every element is in its own conjugacy class of size one because addition is a commutative operation.