Let G be an arbitrary group. Consider the following relations on G:
R1: ∀a, b ∈ G, a R1 b if and only if ∃g ∈ G such that a = g^−1bg
R2: ∀a, b ∈ G, a R2 b if and only if a = b^−1
Which of the above is/are equivalence relation/relations?

Question

Accepted Answer

R1 only. R1 (a = g^-1 b g) is conjugacy, which is reflexive, symmetric, and transitive in any group, so it is an equivalence relation. R2 (a = b^-1) fails reflexivity since a = a^-1 requires a^2 = e, not true for all a. Only R1, index 1, not 'R1 and R2'.

Let G be an arbitrary group. Consider the following relations on G: R1: ∀a, b ∈ G, a R1 b if and only if ∃g ∈ G such that a = g^−1bg R2: ∀a, b ∈ G, a R2 b if and only if a = b^−1 Which of the above is/are equivalence relation/relations?

Solution

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