3.4: Cosets and Lagrage's Theorem

In this section, we'll prove Lagrange's Theorem, a very beautiful statement about the size of the subgroups of a finite group. But to do so,we'll need to learn about cosets.

Recall the Cayley graph for the dihedral group $D_5$ as generated by a flip and a rotation. Notice that the darker blue arrows look like two different 'copies' of $\mathbb{Z}_5$ sitting inside of the dihedral group. Likewise, the light arrow loops look like five copies of $\mathbb{Z}_2$. Both $\mathbb{Z}_5$ and $\mathbb{Z}_2$ are subgroups of $D_5$, generated by the rotation and flip respectively. These 'copies' of the subgroups that we see in the Cayley graph are examples of cosets.

Dihedral group Cayley graph, generated by a flip and rotation. The Cayley graph for the dihedral group with generators given by a flip and a rotation.

Definition 3.3.0: Coset

Let $H$ be a subgroup of $G$, and $H=\{h_1, h_2, h_3, \ldots \}$. Then for any choice of $g\in G$, the coset $gH$ is the set $\{gh_1, gh_2, \ldots \}$.

These are precisely the 'copies' of the subgroups that we saw in $D_5$. The elements of $D_5$ can all be written as $f^ir^j$ with $i\in \{0,1\}$ and $j\in \{0,1,\ldots,4\}$. The rotation subgroup consists of the element $R=\{e, r, r^2, r^3, r^4\}$. Then $R$ has two distinct cosets, $R$ and $fR=\{f, fr, fr^2, fr^3, fr^4\}$. For any $g$ we choose, $gR$ is equal to one of these two cosets! Likewise, if we consider the subgroup $F=\{e, f\}$, there are five distinct cosets given by $r^iF$, where $i\in \{0,1,2,3,4\}$. Notice that the cosets evenly divide up the group; this isn't an accident!

Proposition 3.3.1
Suppose that $H$ is a subgroup of $G$. Let $x, y\in G$. Then either $xH=yH$ or $xH$ and $yH$ have no elements in common.

Proof 3.3.2
Suppose $z\in xH$ and $z\in yH$. In particular, there exist $h_1, h_2\in H$ such that $z=xh_1=yh_2$, and $xh_1h_2^{-1}=y$. We need to show that $xH=yH$, so take any $h_3\in H$ and consider $yh_3\in yH$. Then $yh_3=xh_1h_2^{-1}h_3\in xH$, since $h_1h_2^{-1}h_3\in H$. Thus, if $xH$ and $yH$ share any elements, then they are equal, and if they share no elements, they are tautologically disjoint!

We'll need one more piece of notation.

Definition 3.3.3: Order of a group

The order of a group $G$, written $|G|$, is the number of elements in $G$.

So the order of $R\subset D_5$ is $|R|=5$, and the order of the flip subgroup $|F|=2$. We can now prove Lagrange's Theorem!

Theorem 3.3.4: Lagrange's Theorem
Let $H$ be a subgroup of a finite group $G$. Then $|H|$ divides $|G|$.

Proof 3.3.5: Lagrange's Theorem

Notice that every element of the group $G$ shows up in some coset of $H$: since $e\in H$, we have $g\in gH$ for every $g$. Therefore, every element of the group shows up in exactly one coset of $H$. Also notice that every coset of $H$ has the same number of elements as $H$. (If the size of $gH$ were less than $|H|$, there would be have to be two different elements $h_1, h_2\in H$ with $gh_1=gh_2$. But cancelling the $g$'s gives $h_1=h_2$, a contradiction.) Then the cosets of $H$ break up $G$ evenly into subsets of size $|H|$. Thus, $|H|$ divides $|G|$, as desired.

Exercise 3.3.6

Find all of the subgroups of the permutation group $S_3$ and the dihedral group $D_5$.

Exercise 3.3.7

Find a subgroup of the permutation group $S_4$ with twelve elements. What are it's cosets?

Contributors and Attributions

• Tom Denton (Fields Institute/York University in Toronto)