GATE 2019 CS – Question 20

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 1 mark · Multiple choice

Let $G$ be an arbitrary group. Consider the following relations on $G$:

$R_1$: $\forall a,b\in G$, $a\,R_1\,b$ if and only if $\exists g\in G$ such that $a=g^{-1}bg$

$R_2$: $\forall a,b\in G$, $a\,R_2\,b$ if and only if $a=b^{-1}$

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

$R_1$ and $R_2$
$R_1$ only
$R_2$ only
Neither $R_1$ nor $R_2$

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Correct answer: (B) $R_1$ only

Explanation

$R_1$ (conjugacy) is reflexive ($g=e$), symmetric ($b=gag^{-1}$) and transitive, so it is an equivalence relation. $R_2$ is not reflexive, since $a=a^{-1}$ only for elements of order at most 2.