# Statement(s) > [!NOTE] Equivalence of Left and Right Cosets in Abelian Group > Let $(G,\circ)$ be an [[Groups|abelian group]] and $H$ a [[Subgroup|subgroup]] of $G.$ Then for all $g\in G,$ $gH=Hg$ # Proof **Proof.** Let $g\in G$ and $g\circ h\in gH.$ Then by commutativity of $\circ,$ $g\circ h = h\circ g\in Hg,$ thus $gH \subset Hg.$ Similarly, $Hg \subset gH$ so $gH=Hg.$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography