Advanced Algebra 1

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

Group theory

Basic definitions and properties

Definition: A group is a set $G$ with a binary operation $\cdot$ such that:

1. $\cdot$ is associative
2. $\cdot$ has an identity element
3. Every $g \in G$ has an inverse.

Examples:

1. $\mathbb{Z}$ with operation $+$, denoted $(\mathbb{Z},+)$.
2. $\mathbb{Q}^{\times} \coloneq \mathbb{Q} - {0}$ under multiplication.
3. $GL_n(\mathbb{R}) \coloneq \{n \times n \text{invertible real matrices }\}$ under matrix multiplication (n \geq 1).
4. $S_n \coloneq \{\text{permutations of} \{1,2,3,...,n\}\}$.
5. More generally $S_X = \{\text{bijective functions } G: X \rightarrow X\}$ under composition.
6. $Q_{8} \coloneq$ quaternion group of order 8 $= \{\pm 1, \pm i, \pm j, \pm k\}$ where $i^2=j^2=k^2=-1$ and $ij=k=-ji$.
7. Product group: If $G$ and $H$ are groups then $G \times H$ with $\cdot_G$ and $\cdot_H$ operations is a group under $(g,h)\cdot(g',h') = (g\cdot_G g',h \cdot_H h')$ e.g. $\mathbb{Z} \times S_3$ is a group.

Notation: Usually just write $gh$ for $g\cdot h$, unless ($\cdot$ is $+$) and $ghk$ for $(gh)k = g(hk)$, $g^{-1}$ for the inverse of $g$, $g^n = \overbrace{g \cdots g}^{n \ \text{times}}$ if $n>0$, $g^{-n}= (g^n)^{-1}$ if $n>0$. $g^0 = e_G$ the identity element of $G$.

Also note: $g^m g^n = g^{m+n}$ and $(g^m)^n = g^{mn}$ for all $g \in G$ and $m,n \in \mathbb{Z}$. Note that $g^n h^n \neq (gh)^n$ in general, this only holds if $G$ is abelian.

Cancellation law: $g,h,k \in G$ a group, $gh = gk \implies h=k$, likewise for $hg=kg$.

Subgroups: A subgroup of a group $G$ is a subset $H \subset G$ such that:

1. $e_g \in G$
2. $h,h' \in H \implies hh' \in H$
3. $h \in H \implies h^{-1} \in H$.

Examples: $\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}.$

$Z \subset B \subset GL_n(\mathbb{R})$ where $Z$ are the nonzero $n\times n$ scalar matrices, $B$ are the upper-triangular invertible $n \times n$ matrices.

For any $g \in G$, $\{g^n | n \in \mathbb{Z}\}$ is the cyclic subgroup of $G$ generated by $g$, denoted $\langle g \rangle$.

Example: $G = \mathbb{Z}$, $\langle m \rangle = \{nm | n \in \mathbb{Z}\}$ = multiples of $m$ $\subset \mathbb{Z}$.

$G = G^{\times}$, $\langle 3 \rangle = \{3^n | n \in \mathbb{Z}\}$. $G = S_3$, $\langle (123)\rangle = \{e, (123), (132)\}$.

More generally, for any subset $S \subset G$, define the subgroup of $G$ generated by $S$ to be:

where $A = \{\text{subgroups } H\subset G | S \subset H\}$. This is the smallest subgroup of $G$ containing $S$ (Exercise).

Example: $G = \mathbb{Z}$, $m,n \in \mathbb{Z}$ not both $0$, $\langle \{m,n\} \rangle = \langle d \rangle$ where $d = gcd(m,n)$. $G = S_3$, not cyclic (not even abelian): $S_3 = \langle (12),(123) \rangle$.

$G = G^{\times}$, $\langle 2,3 \rangle = \{2^a3^b | a,b \in \mathbb{Z}\}$ If $G$ is a group then $|G|$ is called the order of $G$ (may be infinite). If $g \in G$ then the order of $g$ is the order of $\langle g \rangle$ (may be infinite).

Examples: $S_n$ has order $n!$. $(12)$ has order 2, $(123)$ has order 3, $(12)(34)$ has order 2.