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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