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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 $6$.