Lagrange’s Theorem
See this video and these notes
Our goal is to prove the following fundamental fact about the orders of finite groups and their subgroups.
Theorem: Let $G$ be a finite group and let $H$ be a subgroup. Then the number of elements (the order) of $H$ is a divisor of the order of $G$.
Corollary: Let $G$ be a finite group and let $g\in G$. Then the order of the element $g$ divides the order of the group $G$.
Proof (of Corollary): The order of $g$ is the number of elements in the cyclic subgroup $\langle g\rangle$ generated by $G$, and, by the theorem, this number is a divisor of the order of $G$.
Cosets
See this video and these notes.
Cosets are the main tool in proving Lagrange’s theorem. Let $G$ be a group and $H$ be a subgroup. For now, we don’t assume these groups are finite.
Definition: Given $g\in G$, let \(gH =\{gh : h\in H\}\) and \(Hg = \{hg : h\in H\}.\) $gH$ is called a left coset of $H$, and $Hg$ is called a right coset of $H$ in $G$.
Proposition: Given $g_1,g_2\in G$, the following are equivalent:
 $g_1H=g_2H$
 $Hg_1^{1}=Hg_2^{1}$
 $g_1H\subset g_2H$
 $g2\in g_1H$
 $g_1^{1}g_2\in H$
Theorem: Let $G$ be a group and $H$ a subgroup. There is a bijection between the sets of left and right cosets of $H$. In particular, if the number of left cosets (or right cosets) is finite, the number of distinct left cosets $gH$ is the same as the number of distinct right cosets $Hk$.
Definition: The index of $H$ in $G$, written $[G:H]$, is the number of distinct (left or right) cosets of $H$ in $G$, if this number is finite. Otherwise we say that $H$ has infinite index in $G$.
Examples

if $G=\mathbb{Z}$ and $H=n\mathbb{Z}$, the cosets of $H$ are the arithmetic progressions \(i+n\mathbb{Z}=\{x\in\mathbb{Z}: x\equiv i\pmod{n}\}\)

if $G=\mathbb{Z}_6$ and \(H=\{0,3\}\) then the cosets are \(\{0,3\}\), \(\{1,4\}\), and \(\{2,5\}\).

if $G=D_{n}$ and $H$ is the subgroup of rotations, then the left (and right) cosets are $H$ and the set $sH$ of reflections.
Proof of Lagrange’s Theorem
See this video and these notes
The proof follows from the following. Suppose $G$ is finite.
 Every (left) coset $gH$ has the same number of elements as $H$. The (left) cosets $gH$ partition $G$, in the sense that the following properties hold:
 Either $g_1H=g_2H$ or $g_1H\cap g_2H=\emptyset$
 $G$ is the union of the distinct $gH$.
Since $G$ is the disjoint union of $[G:H]$ cosets, each with $H$ elements, we have \(G = [G:H]H\) and so $H$ is a divisor of $G$.
The Converse of Lagrange’s Theorem is False
See this video and these notes.
Proposition: The group $A_4$, of order $12$, has no subgroup of order $6$.