Left Coset Equals Subgroup iff Element in Subgroup

Theorem

Let $G$ be a group whose identity is $e$.

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

Let $x \in G$.

Let $x H$ denote the left coset of $H$ by $x$.

Then:

$x H = H \iff x \in H$

$\ds x H$

$=$

$\ds H$

$\ds \leadstoandfrom \ \ $

$\ds x H$

$=$

$\ds e H$

$\ds \leadstoandfrom \ \ $

$\ds x e^{-1}$

$\in$

$\ds H$

$\ds \leadstoandfrom \ \ $

$\ds x$

$\in$

$\ds H$

Group Properties

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $112$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42.6 \ \text {(3L)}$ Another approach to cosets
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $2$