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Cosets and index

Proposition: If $K$ is a subgroup of $H$ and $H$ is a subgroup of $G$ then $[G:K]=[G:H][H:K].$

Proposition: If $H$ and $K$ are subgroups of $G$, and $HK=\lbrace hk : h\in H, k\in K\rbrace$ then

  1. $\mid HK\mid = \frac{\mid H \mid \mid K \mid}{\mid H \cap K \mid}$
  2. $HK$ is a subgroup if and only if $HK=KH$. This holds if $H$ is a subgroup of $N_{G}(K)$, and a fortiori if $K$ is normal in $G$.
  3. If $m$ and $n$ are $[G:H]$ and $[G:K]$ respectively, then the index of $H\cap K$ in $G$ is between $\mathrm{lcm}(m,n)$ and $mn$.

Coset examples

Example 1 Let $G=S_n$. The permutation group $S_k$ is a subgroup of $S_n$. What are its cosets? There are $n!/k!$ of them.

The coset representatives can be viewed as maps from the set $\lbrace k+1,\ldots, n\rbrace$ to the set $\lbrace 1,\ldots, n\rbrace.$

Example 2 Let $G=S_n$. The permutation group $S_k\times S_{n-k}$ is a subgroup of $S_{n}$ where $S_{n-k}$ is viewed as the permutations of the set $\lbrace k+1,\ldots,\rbrace$. What are the cosets of this subgroup?

Example 3 Let $G=\GL_{2}(\mathbb{R})$. Let $P$ be the subgroup of upper triangular matrices. Show that the cosets $G/P$ are in bijection with equivalence classes of vectors $[x,y]$ where $xy\not=0$ and $[x,y]\sim [x’,y’]$ whenever there is a nonzero constant $a$ so that $ax=x’$ and $ay=y’$.

Equivalently show that the cosets are in bijection with the lines through the origin in $\mathbb{R}^{2}$.

Example 4 Let $G=D_{2n}$ and let $H$ be a subgroup generated by a reflection fixing a vertex. Show that the cosets of $H$ are in bijection with the $n$ vertices of the polygon on which $D_{2n}$ acts in its usual representation.

Example 5 Let $G$ be the affine group of $\mathbb{R}^{2}$. Let $H$ be the copy of $\GL_{2}(\mathbb{R})$ inside $G$. Show that the cosets are in bijection with the points of $\mathbb{R}^{2}$.

Quotients

Show that every subgroup of the quaternion group $Q_{8}$ is normal. What are the quotient groups?

Show that every quotient group of a cyclic group is cyclic.

A group is simple if it has no nontrivial normal subgroups, hence no nontrivial quotients. $\Zn{p}$, for $p$ prime, are the simple cyclic groups.

$\SL_{n}(F)$ is the kernel of the determinant map $\GL_{n}(F)\to F^{\times}$.

$\PGL_{n}(F)$ is the quotient of $\GL_{n}(F)$ by the normal subgroup of matrices $aI_{n}$ for $a\in F^{\times}$.

Universal property

  1. Every homomorphism $f:G\to H$ makes a quotient of $G$ into a subgroup of $H$.
  2. Every surjective homomorphism $f:G\to H$ is an isomorphism from a quotient of $G$ to $H$.
  3. Every injective homomorphism $f:G\to H$ makes $G$ into a subgroup of $H$.
  4. If $G$ is simple, every homomorphism $f:G\to H$ is either trivial or injective.

Examples

The group \(S^{1}=\lbrace z\in\C : \|z\|=1\rbrace\) is isomorphic to $\R/\Z$. The projection map is $\theta\mapsto e^{i\theta}$.

The subgroup $\Q/\Z\subset\R/\Z$ is isomorphic to the roots of unity.

Let $D_{2n}$ be the dihedral group of order $2n$ and suppose $d\mid n$. Then $\Zn{d}$ is a normal subgroup of $D_{2n}$ and the quotient group is $D_{2n/d}$

The alternating group

The alternating group $A_{n}$ is the subgroup of $S_{n}$ consisting of even permutations. It is the kernel of the sign homomorphism.