Advanced Algebra 3

This is a compiled set of notes separated by topic on advanced algebra, based on lectures at King's College London.

Group theory: Isomorphism Theorems

Example:

f: \mathbb{R}^{\times} \rightarrow \mathbb{R}^{\times} \\ r \mapsto r^2

has $\ker(f) = \{\pm 1\} = \langle -1 \rangle$ , and $im(f) = \mathbb{R}_{>0}$ . The first isomorpishm theorem says that $\overline{f}: \mathbb{R}^{\times}/\{\pm 1\} \rightarrow \mathbb{R}_{>0}$ is an isomorphism of groups.

Lemma: If $H, N \subset G$ and $N \lhd G$ then $HN = NH$ is a subgroup of $G$ .

Proof:

$HN = NH$ since $hN = Nh \ \forall h$ . Now let $h_1n_1$ and $h_2n_2 \in HN$ . then:

h_1n_1h_2n_2 = h_1h_2h_2^{-1}n_1h_2n_2 \\ = h_1h_2(h_2^{-1}n_1h_2)n_2 \in HN

$\blacksquare$

Remark: If $H,K \subset G$ are subgroups the $HK$ is not necessarily a subgroup, for example $G=S_3,\ H=\langle(12)\rangle, \ K = \langle (23)\rangle$ means $HK = \{e,(12),(23),(123)\}$ which doesn't include $(23)(12) = (132)$ .

Second Isomorphism Theorem: Suppose $H,N \subset G$ are subgroups and $N \lhd G$ . Then:

$H \cap N \lhd H$
$N \lhd HN$
$H/(H\cap N) \rightarrow HN/N$ where $\overline{h} \coloneq h (H \cap N) \mapsto hN$ is an isomorphism.

Proof:

$H \lhd N$ and $N \lhd G$ , meaning $x \in H\cap N$ , $h \in H$ , and $hxh^{-1} \in H$ since $x \in H$ , $hxh^{-1} \in N$ since $x \in N \lhd G$ meaning $hxh^{-1} \in H \cap N$ . This shows $H \cap N \lhd H$ .
This is from $N \lhd G$ .
We have group homomorphisms: $H \hookrightarrow HN \\ HN \twoheadrightarrow HN/N$ This composition is a group hom. $f: H \rightarrow HN/N$ with image $HN/N$ and kernel $\ker(f) = H \cap N$ . By the first isomorphism theorem $\overline{f}: H/(H\cap N) \rightarrow HN/N$ is an isomorphism. $\blacksquare$

Example: $H$ are the diagonal matrices with entries in $\mathbb{R}^{\times}$ . $G = B^+$ are the upper-triangular matrices in $GL_n(\mathbb{R})$ . $N$ are the unipotent upper-triangular matrices in $B^+$ . The second isomorphism theorem says that $H/(H\cap N) \xrightarrow{\sim} HN/N$ meaning $H \xrightarrow{\sim} B^+/N$ .

Group Actions Let $G$ a group $X$ a set. Definition: An action of $G$ on $X$ is a map:

G \times X \rightarrow X\\ (g,x) \mapsto g\cdot X

such that:

$e\cdot x = x$ for all $x \in X$
$g\cdot (h \cdot x) = (gh)\cdot x$ , where $g,h \in G$ and $x \in X$ .

We say $G$ acts on $X$ or $X$ is a $G$ -set.

Examples:

$H \subset G$ , then $X = G/H$ is a $G$ -set via: $G \times G/H \rightarrow G/H \\ (g,xH) \mapsto g \times h$
$G = S_n$ acts on $X = \{1,2,...,n\}$ or $S_X$ acts on $X$ more generally.
$G = D_n$ acts on the set $V$ of vertices of a regular $n$ -gon.
$GL_n(\mathbb{R})$ acts on $\mathbb{R}^n$ by $A \cdot \underline{v} = A\underline{v}$ .
Given any $G$ and any $X$ there is always a trivial action $g\cdot x = x$ for all $g \in G$ and $x \in X$ .
If $G$ acts on sets $X$ and $Y$ then $G$ acts on $X \times Y$ by $g\cdot (x,y) = (g\cdot x, g \cdot y)$ for all $g \in G$ and $x \in X, \ y \in Y$ .
If $H \subset G$ is a subgroup of $G$ and $X$ is a $G$ -set then $H$ acts on $X$ .

Conjugation action of $G$ on $G$ : $g \in G$ and $x \in X = G$ , conjugation defines an action:

g \cdot x = gxg^{-1}

Definition: For $X$ a $G$ -set and $x \in G$ , the orbit of $x$ is $G \cdot x = \{g\cdot x | g \in G\}$ . The stabilizer subgroup (or isotropy subgroup) of $x$ is $G_x = \{g | g \cdot x = x\}$ .

Examples:

Orbit of $x \in X = G$ under conjugation action is the conjugacy class of $x$ .
The stabilizer subgroup of $x \in X = G$ under conjugation is: $G_x = \{g \in G | gxg^{-1} = x\} = Z_G(x)$ the centralizer of $x$ in $G$ .