# Math4302 Modern Algebra (Lecture 3)

## Groups

<details>
<summary>More examples for groups</summary>

Let $\mathbb{Q}^+$ be the set of positive rational numbers.

Then $(\mathbb{Q}^+,\times)$ is a abelian group with identity $1$ and inverse $a^{-1}=\frac{1}{a}$.

If we defined $*$ by $a*b=\frac{ab}{2}$, then we have identity $2$. $a*e=\frac{ae}{2}=a$, we have $e=2$.

and inverse $a^{-1}a=\frac{a^2}{2}=2$, therefore $a^{-1}=\frac{4}{a}$.

This is also an abelian group.

</details>

### Properties for groups

- $(a*b)^{-1}=b^{-1}*a^{-1}$ (inverse)
- $a*b=a*c\implies b=c$ (cancellation on the left)
- $b*a=c*a\implies b=c$ (cancellation on the right)
- If $a*b=e$, then $b=a^{-1}$ (we can solve linear equations)

#### Additional notation

for $n\geq 1$,

- $a^n=a*a\cdot \cdots \cdot a$ (n times)
- $a^{-n}=a^{-1}\cdot \cdots \cdot a^{-1}$ (n times)

for $n=0$, $a^0=e$

We can easily prove this is equivalent to our usual sense for power notations.

That is

- $a^n*a^m=a^{n+m}$
- $(a^n)^m=a^{nm}$
- $a^{-n}=(a^{-1})^n$

### Finite groups

Group with 4 elements.

|*|e|a|b|c|
|---|---|---|---|---|
|e|e|a|b|c|
|a|a|b|c|e|
|b|b|c|e|a|
|c|c|e|a|b|

Note $a,c$ are inverses and $b$ self inverse.

_isomorphic to $(\mathbb{Z}_4,+)$, $(\{1,-1,i,-i\},\cdot)$_

and we may also have

|*|e|a|b|c|
|---|---|---|---|---|
|e|e|a|b|c|
|a|a|e|c|b|
|b|b|c|e|a|
|c|c|b|a|e|

is

#### Cyclic groups

It is the group of integers modulo addition $n$.

- Associativity: $(a+b)+c=a+(b+c)$
- Identity: $a+0=a$
- Inverses: $a+(-a)=0$

For group with $4$ elements

|*|0|1|2|3|
|---|---|---|---|---|
|0|0|1|2|3|
|1|1|2|3|0|
|2|2|3|0|1|
|3|3|0|1|2|

#### Complex numbers

Consider $\{1,i,-1,-i\}$ with multiplication.

|*|1|i|-1|-i|
|---|---|---|---|---|
|1|1|i|-1|-i|
|i|i|-1|-i|1|
|-1|-1|-i|1|i|
|-i|-i|1|i|-1|

Note that if we replace $1$ with $0$ and $i$ with $1$, and $-1$ with $2$ and $-i$ with $3$, you get the exact the same table as $\mathbb{Z}_4$.