The isomorphism theorems

Theorem: (See DF Theorem 3.16) Let $f:G\to K$ be a homomorphism of groups, let $N$ be the kernel of $f$, and let $\pi:G\to G/N$ be the canonical projection. Then there is a unique injective homomorphism $\overline{f}:G/N\to K$ such that $\overline{f}\circ\pi=f$.

$\begin{xy} \xymatrix { G\ar[rd]^{f}\ar[d]^{\pi} & \\ G/N\ar[r]_{\overline{f}} & K\\ } \end{xy}$

We sometimes say that “$f$ factors through $\pi$” or “$f$ factors through $G/N$”.

Theorem: (See DF Theorem 3.18) Suppose that $G$ is a group and $A$ and $B$ are subgroups of $G$. Suppose further that $A$ is a subgroup of $N_{G}(B)$ so that $AB$ is a subgroup of $G$. Then

1. $B$ is normal in $AB$.
2. $A\cap B$ is normal in $A$.
3. $AB/B$ is isomorphic to $A/(A\cap B)$.
$\begin{xy} \xymatrix { & G & \\ & AB\ar[u] & \\ A\ar[ur] && B\ar[ul]^{=} \\ & A\cap B\ar[ul]^{=}\ar[ur] & \\ &1\ar[u]&\\ }\end{xy}$

The arrows marked with “=” are inclusions of normal subgroups, and the corresponding quotients are isomorphic.

Theorem: (See DF, THeorem 3.19) Let $G$ be a group, and suppose that $H$ and $K$ are normal subgroups of $G$ and $H$ is normal in $K$. Then $K/H$ is normal in $G/H$ and $(G/H)/(K/H)$ is isomorphic to $G/K$.

Theorem: (See DF, Theorem 3.20) Let $G$ be a group and $N$ be a normal subgroup of $G$. Then map $A\mapsto A/N$ is a bijection between the set of all subgroups of $G/N$ and the set of subgroups of $G$ containing $N$. Furthermore, if $A$ and $B$ are subgroups of $G$ containing $N$, then

1. $A\subset B$ if and only if $A/N\subset B/N$
2. If $A\subset B$ then $[A:B]=[A/N:B/N]$
3. $\langle A, B\rangle/N=\langle A/N,B/N\rangle$
4. $(A\cap B)/N=(A/N\cap B/N)$
5. $A$ is normal in $G$ if and only if $A/N$ is normal in $G/N$.

In other words, the lattice of subgroups of $G/N$ is exactly the sublattice of the lattice of subgroups of $G$ containing $N$.

Group Actions

Definition: Let $G$ be a group and $X$ be a set. A action of $G$ on $X$ is a map $$a: G\times X\to X$$ that satisfies $a(e,x)=x$ for all $x$ and $a(g,a(h,x))=a(gh,x)$ for all $g,h\in G$ and $x\in X$. (Remark: We usually write $gx$ or $g\cdot x$ instead of referring to the map $a$)

Equivalently, an action of $G$ on $X$ is a homomorphism $f:G\to S(X)$.

Note: Whenever we have a function $f: A\times B\to C$ we can think of it equivalently as a function $f:A\to \mathcal{F}(B,C)$ where $\mathcal{F}(B,C)$ is the set of all functions from $B$ to $C$. The point is that we can take our function $f:A\times B\to C$, which is a function of two variables, and define $\tilde{f}:A\to\mathcal{F}(B,C)$ by definining $\tilde{f}(a)$ to be the function $\tilde{f}(a)(b)=f(a,b)$. Conversely, if $h:A\to\mathcal{F}(B,C)$ is a function, we can make a function $\overline{h}:A\times B\to C$ by setting $\overline{h}(a,b)=h(a)(b)$. These are mutually inverse constructions so $\mathcal{F}(A\times B,C)=\mathcal{F}(A,\mathcal{F}(B,C))$. This is a property of the cartesian product called adjointness or more specifically left adjointness.

Key Terminology

1. Let $x\in X$. The set $Gx=\lbrace gx: g\in G\rbrace\subset X$ is called the orbit of $x$. More generally, if $H$ is a subgroup of $G$ then $Hx$, defined similarly, is the orbit of $x$ under $H$.
2. Let $x\in X$. The set $\mathrm{Stab}_{G}(x)=\lbrace g : gx=x\rbrace\subset G$ is called the stabilizer of $x$. It is a subgroup of $G$.
3. An action is called transitive if there is an $x\in X$ so that $X=Gx$.
4. The set of $g\in G$ such that $gx=x$ for all $x\in X$ is the kernel of the action.
5. An action is faithful if its kernel is trivial; in other words, if the corresponding map $G\to S(X)$ is injective.
6. If $G$ acts on $X$ and $Y$, a map $f:X\to Y$ is called a morphism of actions if $f(gx)=gf(x)$. If $f$ is bijective then it is an isomorphism of actions.

Key formalities

1. If $G$ acts on $X$, the action partitions $X$ into a disjoint union of orbits. These can be seen as the equivalence classes for the equivalence relation $x\sim y\iff x=gy$ for some $g\in G$.
2. The action of $G$ on each orbit is transitive (by definition).
3. Given $x\in X$, the map $$\pi: gH\mapsto gx$$ gives a well-defined bijection between the cosets of $H=G/\mathrm{Stab}_{G}(x)$ and the orbit $Gx$.
This bijection is an isomorphism of group actions, since if $H= \Stab_G(x)$, then $k\pi(gH)=kgx=\pi(kgH)$.

If $G$ is finite, the size of each orbit is a divisor of the order of $G$.

Key examples

1. If $G$ is a group, and $H$ is a subgroup, let $X$ be the set of left cosets of $H$ in $G$ (regardless of whether $H$ is normal). Then $G$ acts on $X$ via $g\cdot kH=gkH$. The set $X$ is called a homogeneous space for $G$ and is sometimes written $G/H$ even when $H$ isn’t normal. Property 3 under “formalities” says that every orbit in a group action is isomorphic to a homogeneous space for the group. Notice that if $H$ is the trivial subgroup, then this is the action of $G$ on itself by left multiplication; this is called the (left) regular action.+-
2. If $G$ is a group, then $G$ acts on itself via conjugation: $g\cdot h=ghg^{-1}$. The orbits are called conjugacy classes. The stabilizer of an element $g$ under conjugation is the centralizer $C_{G}(\lbrace g \rbrace)$ and the index of this stabilizer is the size of the conjugacy class of $g$.
3. If $g\in Z(G)$ is an element of the center of $G$, then it forms a one-element conjugacy class and its centralizer is all of $G$.

The class equation

Theorem: Let $G$ be a finite group. Let $G$ act on itself by conjugation, yielding a partition of $G$ into disjoint conjugacy classes $K_1,\ldots, K_g$. Choose a representative $g_{i}$ for each class. Then

$\mid G\mid = \sum_{i=1}^{g}\mid K_i\mid = \sum_{i=1}^{g} [G:C_{G}(g_{i})].$

Grouping the conjugacy classes of size one together, we can rewrite this as

$\mid G\mid = \mid Z(G)\mid + \sum_{\lbrace i : \mid K_{i}\mid>1\rbrace} [G:C_{G}(g_{i})]$

This is called the class equation.

Automorphisms

If $G$ is a group, the automorphism group $\Aut(G)$ of $G$ is the set of isomorphisms $G\to G$, with group operation given by composition of functions.

If $G=\Zn{n}$ then $\Aut(G)$ is $(\Zn{n})^{*}$, the multiplicative group of elements mod $n$ that are relatively prime to $n$.

If $G=\Zn{n}^{k}$, then $\Aut(G)$ is $\GL_{n}(\Zn{n})$, the group of $n\times n$ matrices with entries in $\Zn{n}$ that are invertible (meaning their determinant is relatively prime to $n$).

For $g\in G$, conjugation by $g$ is an automorphism of $G$. This gives a homomorphism $G\to \Aut(G)$. THe kernel of this map is the center of $G$. The image is called the group of inner automorphisms. The inner automorphisms form a normal subgroup of the automorphism group.

A group $G$ acts on a normal subgroup $H$ by conjugation. The centralizer of $H$ is the kernel of the action. Therefore $G/C_{G}(H)$ is a subgroup of $\Aut(H)$. And $G/Z(G)$ is a subgroup of $\Aut(G)$.

Definition: A subgroup $H$ of $G$ is called characteristic if it is fixed by every automorphism of $G$, not just the inner ones.

Weird fact: Every automorphism of $S_{n}$ is inner, except $S_{6}$ has an outer automorphism.