Cosets

Let $G$ be a group and $H$ a subgroup of $G$. For $g$ belonging to $G$, the subset

$$g*H=gH=\{gh\ \vert\ h\in H\},$$

of $G$ is called a left coset of $H$ in $G$ and the subset

$$H*g=Hg=\{hg\ \vert\ h\in H\},$$

of $G$ is called a right coset of $H$ in $G$.

Example 5.13.1   If $H = \langle (1,3),(1,4)\rangle$ and $G=S_4$ as in Example 5.9.2, then $(1, 2, 3)H=\{(1,2,3),(1,2,3)(1,3), (1,2,3)(1,4),...\}$ ( $=(1,2,3)H=\{(1,2,3),(1,2),(1,2,3,4),...\}$) and $H(1,2,3)=\{(1,2,3),(1,3)(1,2,3),(1,4)(1,2,3),...\}$ ( $=\{(1,2,3),(2,3),...\}$). Does $(1,2,3)H=H(1,2,3)$?

Notation: The set of all left cosets is written $G/H$ and the set of all right cosets of $H$ in $G$ is denoted $H\backslash G$.

These two sets don't in general inherit a group structure from G but they are useful none-the-less. ($G/H$ is a group with the ``obvious" multiplication $(g_1*H)*(g_2*H)=(g_1g_2)*H$ if and only if $H$ is a ``normal" subgroup of $G$ - we will define ``normal" below.)

Let $X$ be a finite set and let $G$ be a finite subgroup of the symmetric group $S_X$. The elements of $G$ permute $X$. With this permutation in mind, we say that $G$ acts on $X$. For $x \in X$ and $g\in G$, let $g*x$ denote $g(x)\in X$. This action satisfies the following properties:

• for all $g\in G$, $g:X\rightarrow X$ is a bijection,
• for all $x \in X$, $1*x=x$,
• for all $g,h\in G$, $(gh)*x=g*(h*x)$.

We call the set of images

$$G*x=\{g*x\ \vert\ g\in G\}$$

the orbit of $x$ under $G$. For each $x,y\in X$, we define $x\sim y$ if and only if $y=g*x$, for some $g\in G$. We leave it as an exercise to verify that $\sim$ is an equivalence relation on $X$, in the sense of §1.7.2. The equivalence class of $x$ is the orbit of $x$ under $G$: $[x]=G*x=\{g*x\ \vert\ g\in G\}$. The verification of this is also left as an exercise.

We call

$$stab_G(x)=\{g\in G\ \vert\ g*x=x\}$$ (5.3)

the stabilizer of $x$ in $G$. As an example of their usefulness, we have the following relationship between the orbits and the cosets of the stabilizers.

Proposition 5.13.2   Let $G$ be a finite group acting on a set $X$. Then

$$\vert G*x\vert = \vert G/stab_G(x)\vert,$$

for all $x$ belonging to $X$.

proof: The map

$$g*stab_G(x) \longmapsto g*x$$

defines a function $f : G/stab_G(x) \rightarrow G*x$. The interested reader can easily check that this function is a bijection since it is both an injection and a surjection. $\Box$

Corollary 5.13.3   Let $G$ be a finite group acting on itself by conjugation. Let $S\subset G$ denote a complete set of representatives of the conjugacy classes $G_*$ in $G$ and let $S'=S-Z(G)$, i.e., the subset of $S$ of those elements which are not central. Then

$$G = \cup_{x\in S} Cl(x) =\cup_{x\in S}G/stab_G(x) =Z(G)\cup \cup_{x\in S'}G/stab_G(x),$$

for all $x$ belonging to $X$. In particular,

$$\vert G\vert = \sum_{x\in S} \vert G/stab_G(x)\vert =\vert Z(G)\vert+\sum_{x\in S'} \vert G/stab_G(x)\vert,$$

proof: In the first displayed equation: The first equation is Theorem 5.12.5. The second equality follows from the above proposition.

By taking cardinalities, the second displayed equation is a consequence of the first. $\Box$

Corollary 5.13.4   Theorem 5.8.7(a) holds.

proof: The argument is by induction on $\vert G\vert$.

The result is trivial if $\vert G\vert=1$ (since then no prime divides $\vert G\vert$).

Suppose $\vert G\vert>1$ and let $p$ be a prime dividing $\vert G\vert$. By the induction hypothesis, we assume that the result is true for all subgroups $H$ of $G$ with $\vert H\vert<\vert G\vert$.

Suppose $x\in G$ is not central. Then its centralizer $C_G(x)$ is a proper subgroup of $G$. If $p\vert\vert C_G(x)\vert$ then the result follows from the induction hypothesis. If $p$ does not divide $\vert C_G(x)\vert$ (for all non-central $x$) then since $\vert G\vert=\vert C_G(x)\vert\vert G/stab_G(x)\vert$, by Proposition 5.13.2, it follows that $p\vert\vert G/stab_G(x)\vert$. By Corollary 5.13.3 above, we must have $p\vert\vert Z(G)\vert$. If $Z(G)$ is a proper subgroup of $G$ then we are done by the induction hypothesis.

Thus we may assume $G=Z(G)$ is abelian. If $G$ is cyclic then we leave it to the reader to show that $G$ contains an element of order $p$. We shall assume that $G$ is not cyclic. Let $H$ be a proper subgroup of $G$ of maximal order (this exists since $G$ is not cyclic) and let $a\in G-H$. Then $\langle a\rangle \cap H=\{1\}$ (else $a\in H$) and $G=H\cdot \langle a\rangle$ (else $H$ would not be maximal). Thus $p$ either divides $H$ or $\vert\langle a\rangle \vert$. In either case, the result follows from the induction hypothesis. $\Box$

Theorem 5.13.5 (Lagrange)   : If $G$ is a finite group and $H$ a subgroup then

$$\vert G/H\vert = \vert G\vert/\vert H\vert.$$

Corollary 5.13.6   If $H, G$ are as above then the order of $H$ divides the order of $G$.

Now we prove the Theorem.

proof: Let $X$ be the set of left cosets of $H$ in $G$ and let $G$ act on $X$ by left multiplication. Apply the previous lemma with $x=H$. $\Box$

Definition 5.13.7   : Let $H$ be a subgroup of $G$ and let $C$ be a left coset of $H$ in $G$. We call an element $g$ of $G$ a coset representative of $C$ if $C = g*H$. A complete set of coset representatives is a subset of $G$, $x_1, x_2, ..., x_m$, such that

$$G/H = \{x_1*H, ..., x_m*H \},$$

without repetition (i.e., all the $x_i*H$ are disjoint).

Exercise 5.13.8   Let $G$ be the group of symmetries of the square. Using the notation above, compute $G/\langle g_3\rangle$ and $G*x_0$.

Exercise 5.13.9   For $g_1,g_2 \in G$, define $g_1 \sim g_2$ if and only if $g_1$ and $g_2$ belong to the same left coset of $H$ in $G$.

(a) Show that $\sim$ is an equivalence relation.

(b) Show that the left cosets of $H$ in $G$ partition $G$.

Exercise 5.13.10   If $H$ is finite, show $\vert H\vert = \vert g*H\vert = \vert H*g\vert$.

Exercise 5.13.11   Let $G=\mathbb{Z}$ and $H=3\mathbb{Z}$ (i.e. $H$ is the set of multiples of 3).

(a) Prove that $H\ <\ G$.

(b) What is $\vert\mathbb{Z}/3\mathbb{Z}\vert$?

Exercise 5.13.12   Suppose $\vert G\vert<30$ and $G$ is nonabelian and $g\in G$ is an element of order $11$.

(a) What is $\vert G\vert$?

(b) What is $G$ and what is $g$?

Exercise 5.13.13   Let $G$ be the symmetry group of a cube. By this, we mean the following: If $X$ is the set of vertices of a cube, e.g. $\{(0,0,0),...,(1,1,1)\}$, consider the subgroup of the full permutation group of $X$ that arises from physically possible rotations of space. (This is the analog of the symmetry group of the square, $D_8$, discussed earlier.)

(a) $G$ can be considered to be a subgroup of $S_8$. Find $G$, $\vert G\vert$ and $\vert S_8/G\vert$.

(b) What is $\vert stab_G((1,1,1))\vert$?

Exercise 5.13.14   If $X$ is a left coset of $H$ in $G$ and $x$ is an element of $G$, show that $x*X$ is also a left coset of $H$ in $G$.

Exercise 5.13.15   Let $G=S_3$, the symmetric group on 3 letters, and let $H = \langle s_1\rangle$, in the notation of §5.3 above.

(a) Compute $\vert G/H\vert$ using Lagrange's Theorem.

(b) Explicitly write down all the cosets of $H$ in $G$.