Let G be an arbitrary group. Consider the following relations on G:
R1: For all a, b in G, a R1 b if and only if there exists g in G such that a = g^(-1) * b * g
R2: For all a, b in G, a R2 b 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
GATE 2019 · Discrete Mathematics · Group Theory · medium
Answer: Only R1 is an equivalence relation, so the answer is B.
Check R1 (conjugacy): Reflexive: a = e^-1 a e. Symmetric: a = g^-1 b g => b = (g^-1)^-1 a (g^-1). Transitive: a = g^-1 b g, b = h^-1 c h => a = (hg)^-1 c (hg). All three hold.
Check R2 (a = b^-1): Reflexivity needs a = a^-1, i.e. a^2 = e, which fails for any element of order > 2 (e.g. in Z_4). So R2 is not reflexive and hence not an equivalence relation.