Coset

Given a subgroup $H$ of a group $G$ and an element $g \in G$, the left coset of $H$ by $g$ is $gH = \{g \cdot h \mid h \in H\}$. The right coset is $Hg = \{h \cdot g \mid h \in H\}$.

Cosets partition $G$ into disjoint subsets of equal size. Two left cosets $gH$ and $g'H$ are either identical or disjoint. This partition gives Lagrange’s theorem: $|G| = |G : H| \cdot |H|$, where $|G : H|$ is the number of distinct cosets (the index of $H$ in $G$).

When every left coset equals the corresponding right coset ($gH = Hg$ for all $g$), the subgroup is normal and the cosets form a quotient group.