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

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

𝑅₁: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅₁𝑏 if and only if ∃𝑔 ∈ 𝐺 such that 𝑎 = 𝑔⁻¹𝑏𝑔

𝑅₂: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅₂𝑏 if and only if 𝑎 = 𝑏⁻¹

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

(A) 𝑅₁ and 𝑅₂

(B) 𝑅₁ only

(C) 𝑅₂ only

(D) Neither 𝑅₁ nor 𝑅₂

Ans: 𝑅₁ only

Solution: