ThmDex – An index of mathematical definitions, results, and conjectures.

Result R738 on D23: Abelian group

Left and right cosets coincide in Abelian group

Formulation 0

Let $G$ form an D23: Abelian group.
Let $H$ form a D496: Subgroup of $G$.

Then \begin{equation} \forall \, g \in G : g H = H g \end{equation}

Proofs

Proof 0

Let $G$ form an D23: Abelian group.
Let $H$ form a D496: Subgroup of $G$.

Applying commutativity yields \begin{equation} g H : = \{ g h \in h \in H \} = \{ h g \mid h \in H \} = : H g \end{equation} $\square$