# 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

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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