Question

Let G be an arbitrary group. Consider the following relations on G:

R1: ∀a,b∈G, a R1 b if and only if ∃g∈G such that a=g⁻¹bg
R2: ∀a,b∈G, a R2 b if and only if a=b⁻¹

Which of the above is/are equivalence relation/relations?

Accepted Answer

The correct answer is (B) R1 only. R1 is the standard CONJUGACY relation, always an equivalence relation for any group (reflexive via g=e, symmetric and transitive follow from group axioms). R2 (a=b⁻¹) is NOT reflexive in general — it would require a=a⁻¹ for every a, which fails whenever G has an element of order >2. So only R1 is an equivalence relation.

Explanation