Relations on groups.

Question

Let $G$ be a group and $H$ a subgroup of $G$. Define a relation on elements of $G$ by saying that $a \sim b$ if $b^{-1} a \in H$. This relation is :

• a) reflexive and symmetric, but transitive only if G is abelian.
• b) reflexive and transitive, but symmetric only if G is abelian.
• c) reflexive, symmetric and transitive.

My instinct would be to answer b) because abelian implies commutativity. However, I may be wrong but I don’t think it’s a).

Thank you.

Answer 1

Note that $a \sim b$ iff $aH=bH$ (*). This should make it easy to answer the question.

In general, every relation on a set $X$ defined by the equality of values of a function on $X$ must be an equivalence relation because equality is an equivalence relation.

Here, $X=G$ and the function is $a \mapsto aH$.

(*) Indeed:

$a \sim b \implies aH = eaH = (bb^{-1})aH = b(b^{-1}a)H = b(b^{-1}aH)=bH$, because $b^{-1}a\in H$.

$aH=bH \implies a=ae=bh \implies b^{-1}a=h\in H$.

Answer 2

HINT

$a^{-1}a = 1 \in H$, so $\sim$ is reflexive

$a \sim b \rightarrow b^{-1}a \in H \rightarrow \left(b^{-1} a\right)^{-1} = a^{-1} b \in H \rightarrow b \sim a$ so it is symmetric

can you check transitivity?