Congruence Class Modulo Subgroup is Coset

Theorem

Let $G$ be a group, and let $H \le G$.

Left Congruence Class

Let $\mathcal R^l_H$ be the equivalence defined as left congruence modulo $H$.

The equivalence class $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$ of an element $g \in G$ is the left coset $g H$.

This is known as the left congruence class of $g \bmod H$.

Right Congruence Class

Similarly, let $\mathcal R^r_H$ be the equivalence defined as right congruence modulo $H$.

The equivalence class $\left[\!\left[{g}\right]\!\right]_{\mathcal R^r_H}$ of an element $g \in G$ is the right coset $H g$.

This is known as the right congruence class of $g \bmod H$.

Coset Spaces form Partition

The left coset space of $H$ forms a partition of its group $G$, and hence:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \ \equiv^l y \ \left({\bmod H}\right)\)	\(\iff\)	\(\displaystyle x H = y H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \neg \ \left({x \ \equiv^l \ y}\right) \ \left({\bmod H}\right)\)	\(\iff\)	\(\displaystyle x H \cap y H = \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Similarly, the right coset space of $H$ forms a partition of its group $G$:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \ \equiv^r y \ \left({\bmod H}\right)\)	\(\iff\)	\(\displaystyle H x = H y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \neg \ \left({x \ \equiv^r \ y}\right) \ \left({\bmod H}\right)\)	\(\iff\)	\(\displaystyle H x \cap H y = \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Uniqueness of Cosets

Hence:

• For each $x \in G$ there exists a unique left coset of $H$ containing $x$, that is: $x H$
• For each $x \in G$ there exists a unique right coset of $H$ containing $x$, that is: $H x$.

Proof

Proof of Left Congruence Class

• Let $x \in \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x \in \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \exists h \in H: g^{-1} x = h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of left congruence modulo $H$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \exists h \in H: x = g h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group properties
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in g H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Left Coset
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H} \subseteq g H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Subgroup

• Now let $x \in g H$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle x \in g H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle \exists h \in H: x = g h\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Left Coset
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle g^{-1} x = h \in H\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Group properties
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of left congruence modulo $H$
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle g H \subseteq \left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Subgroup

Thus $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H} = g H$, that is, the equivalence class $\left[\!\left[{g}\right]\!\right]_{\mathcal R^l_H}$ of an element $g \in G$ equals the left coset $g H$.

$\blacksquare$

Proof of Right Congruence Class

The proof for this follows the same structure as the proof for the Left Congruence Class.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 11$: Theorem $11.1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 37$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 42.3$
• John F. Humphreys: A Course in Group Theory (1996): $\S 5$: Proposition $5.4$