Proof of Left Cosets Partition a Group and All Have the Same Cardinality

Throughout, the operation of $G$ is associative with neutral element $e$ (Groups and Abelian Groups (i), Monoids (i)), the subgroup $H$ contains $e$ and is closed under products and inverses (Subgroups (i)), and left cosets are as in Cosets and the Index of a Subgroup (i); routine facts about inclusions, disjoint classes and the relation of lying in a common left coset are used without comment (Toolkit for AL1).

We first prove (ii). Suppose $gH=kH$. Then $k=ke\in kH=gH$, so $k=gh_{0}$ for some $h_{0}\in H$, and $g^{-1}k=h_{0}\in H$. Conversely, suppose $g^{-1}k=h_{0}\in H$. Then $k=gh_{0}$ and $g=kh_{0}^{-1}$, so for every $h\in H$

$$kh=g(h_{0}h)\in gH\qquad\text{and}\qquad gh=k(h_{0}^{-1}h)\in kH.$$

Hence $kH\subseteq gH$ and $gH\subseteq kH$, that is, $gH=kH$.

For (i), note first that $g=ge\in gH$ for every $g\in G$. Now let $gH$ and $kH$ be two left cosets that are not disjoint, and choose $x\in gH\cap kH$, say $x=gh_{1}=kh_{2}$ with $h_{1},h_{2}\in H$. Then $g^{-1}k=h_{1}h_{2}^{-1}\in H$, so $gH=kH$ by (ii). Thus any two left cosets are equal or disjoint.

For (iii), fix $g\in G$ and consider the function $\lambda:H\to gH$, $\lambda(h)=gh$. It is surjective by the definition of $gH$, and it is injective, because $gh=gh'$ gives

$$h=g^{-1}(gh)=g^{-1}(gh')=h'.$$

So $\lambda$ is a bijection from $H$ onto $gH$ (Injective, Surjective and Bijective Functions between Classes (iii)). If $H$ is finite, then $gH=\lambda(H)$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vi), and Counting: Intervals, Empty Sets and Singletons, Injective Images, Subsets, Unions and Products (iii), applied to $\lambda$ regarded as a function into $G$, gives $\#(gH)=\#H$.

Finally we prove (iv). The relation $\sim$ is reflexive, since $g^{-1}g=e\in H$. It is symmetric: if $g^{-1}k\in H$, then by Cancellation Laws and Basic Inverse Identities in a Group (iv) and Cancellation Laws and Basic Inverse Identities in a Group (iii)

$$(g^{-1}k)^{-1}=k^{-1}g,$$

and this lies in $H$, being the inverse of an element of $H$ (Subgroups (i)). It is transitive: if $g^{-1}k\in H$ and $k^{-1}l\in H$, then $g^{-1}l=(g^{-1}k)(k^{-1}l)\in H$. So $\sim$ is an equivalence relation on $G$ (Equivalence Relations on a Set (i), with Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set (i), Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set (ii) and Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set (iv)). For $g,k\in G$, the element $k$ lies in the equivalence class of $g$ (The Equivalence Class of an Element under an Equivalence Relation (i)) if and only if $g^{-1}k\in H$, that is, $k=gh$ for some $h\in H$, that is, $k\in gH$; so the class of $g$ is $gH$. Hence the quotient of $G$ by $\sim$ (The Quotient of a Set by an Equivalence Relation and the Canonical Projection (i)) is

$$\{[g]_{\sim}: g\in G\}=\{gH: g\in G\}=G/H$$

by Cosets and the Index of a Subgroup (iii).