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Question

Let G be an arbitrary group. Consider the following relations on G:
R1 : a,bϵG,aR1b if and only if gϵG such that a=g1bg
R2: a,bϵG,aR2b if and only if a=b1

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

A
R1 and R2
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B
R1 only
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C
R2 only
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D
Neither R1 nor R2
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Solution

The correct option is B R1 only
R1:a,bϵG,aR1b if and only if gϵG such that a=g1bg
Reflexive: a=g1ag can be satisfied by putting g=e, identity "e" always exists in a group.
So reflexive.
Symmetric: aRba=g1bg for some gb=gag1=(g1)1ag1
g1 always exists for every gϵG
So symmetric
Transitive: aRb and bRca=g11bg1 at b = g12cg2 for some g1g2ϵG Now a=g11g12cg2g1=(g2g1)1cg2g1g1ϵGandg2ϵGg2g1ϵG since group closed so aRb and aRbaRc hence transitive
Clearly R1 is equivalence relation.
R2 is not equivalence it need not even be reflexive since aR2aa=a1a which not be true it is group.
R1 is equivalence relation is the correct answer.

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