Element of Group is in its own Coset/Left

Theorem

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Let $x \in G$.

Let:

$x H$ be the left coset of $x$ modulo $H$.

Then:

$x \in x H$

Proof

Let $e$ be the identity of $G$.

Then:

\(\ds e\)

\(\in\)

\(\ds H\)

Identity of Subgroup

\(\ds x\)

\(=\)

\(\ds x e\)

Definition of Identity Element

\(\ds \leadsto \ \ \)

\(\ds \exists h \in H: \, \)

\(\ds x\)

\(=\)

\(\ds x h\)

Existential Generalisation

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds x H\)

Definition of Left Coset

$\blacksquare$

Also see

• Element of Group is in its own Right Coset

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.5$ Another approach to cosets
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Definition $5.1$: Remark $2$