GATE | GATE CS 2019 | Question 17

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

• R₁: ∀a, b ∈ G, aR₁b if and only if ∃g ∈ G such that a = g⁻¹bg
• R₂: ∀a, b ∈ G, aR₂b 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

Answer: (B)
Explanation: Given R₁ is a equivalence relation, because it satisfied reflexive, symmetric, and transitive conditions:

• Reflexive: a = g^–1ag can be satisfied by putting g = e, identity “e” always exists in a group.
• Symmetric:

  aRb ⇒ a = g–1bg for some g
  ⇒ b = gag–1 = (g–1)–1ag–1
  g–1 always exists for every g ∈ G. 

• Transitive:

  aRb and bRc ⇒ a = g1–1bg1 
  and b = g2–1 cg2 for some g1g2 ∈ G.
  Now a = g1–1 g2–1 cg2g1 = (g2g1)–1 cg2g1
  g1 ∈ G and g2 ∈ G ⇒ g2g1 ∈ G 
  since group is closed so aRb and aRb ⇒ aRc
  
  

R₂ is not equivalence because it does not satisfied reflexive condition of equivalence relation:

aR2a ⇒ a = a–1 ∀a which not be true in a group. 

So, option (B) is correct.
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