Is it true or false that defining a relation on the group $G$ based on the definition $a \sim b$ if and only if $a^{-1}b\in H$ defines an equivalence relation regardless of whether $H$ is a normal subgroup of $G$?
2026-03-26 01:25:16.1774488316
For a group $G$ and subgroup $H$, is $a \sim b \iff a^{-1}b\in H$ an equivalence relation even when $H$ is not normal?
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You can test it yourself. Let $H$ be an arbitrary subgroup of $G$.
Reflexivity: Is it true that $a^{-1}a=1_G\in H$ for each $a\in G$?
Symmetry: Suppose that $a,b\in G$, and $a^{-1}b\in H$; is it true that $b^{-1}a\in H$? HINT: What is $\left(a^{-1}b\right)^{-1}$?
Transitivity: Suppose that $a,b,c\in G$, $a^{-1}b\in H$, and $b^{-1}c\in H$. Can you rewrite $a^{-1}c$ in a way that shows clearly that it’s in $H$?