coset

Definition

Suppose a group $G$, with subgroup $H$, then, a left coset $xH = \{xh : h \in H\}$ for an element $a$ in $G$. Similarly, a right coset may take the form $Ha$.

Notes

• cosets of $H$ in $G$ are equivalence classes for congruence relation,
  • $a \equiv b$ if $b=ah$ for some $h$ in $H$

References

1. M. Artin, Algebra, 2. ed. Boston, Mass. Munich: Pearson Education, Prentice Hall, 2011, p. 56.
2. https://ncatlab.org/nlab/show/coset
3. https://mathworld.wolfram.com/Coset.html