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+# 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 $\mathbb{Z}_4$.
+