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100 lines
2.0 KiB
Markdown
100 lines
2.0 KiB
Markdown
# Math4302 Modern Algebra (Lecture 3)
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## Groups
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<details>
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<summary>More examples for groups</summary>
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Let $\mathbb{Q}^+$ be the set of positive rational numbers.
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Then $(\mathbb{Q}^+,\times)$ is a abelian group with identity $1$ and inverse $a^{-1}=\frac{1}{a}$.
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If we defined $*$ by $a*b=\frac{ab}{2}$, then we have identity $2$. $a*e=\frac{ae}{2}=a$, we have $e=2$.
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and inverse $a^{-1}a=\frac{a^2}{2}=2$, therefore $a^{-1}=\frac{4}{a}$.
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This is also an abelian group.
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</details>
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### Properties for groups
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- $(a*b)^{-1}=b^{-1}*a^{-1}$ (inverse)
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- $a*b=a*c\implies b=c$ (cancellation on the left)
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- $b*a=c*a\implies b=c$ (cancellation on the right)
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- If $a*b=e$, then $b=a^{-1}$ (we can solve linear equations)
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#### Additional notation
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for $n\geq 1$,
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- $a^n=a*a\cdot \cdots \cdot a$ (n times)
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- $a^{-n}=a^{-1}\cdot \cdots \cdot a^{-1}$ (n times)
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for $n=0$, $a^0=e$
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We can easily prove this is equivalent to our usual sense for power notations.
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That is
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- $a^n*a^m=a^{n+m}$
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- $(a^n)^m=a^{nm}$
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- $a^{-n}=(a^{-1})^n$
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### Finite groups
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Group with 4 elements.
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|*|e|a|b|c|
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|---|---|---|---|---|
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|e|e|a|b|c|
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|a|a|b|c|e|
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|b|b|c|e|a|
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|c|c|e|a|b|
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Note $a,c$ are inverses and $b$ self inverse.
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_isomorphic to $(\mathbb{Z}_4,+)$, $(\{1,-1,i,-i\},\cdot)$_
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and we may also have
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|*|e|a|b|c|
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|---|---|---|---|---|
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|e|e|a|b|c|
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|a|a|e|c|b|
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|b|b|c|e|a|
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|c|c|b|a|e|
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is
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#### Cyclic groups
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It is the group of integers modulo addition $n$.
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- Associativity: $(a+b)+c=a+(b+c)$
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- Identity: $a+0=a$
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- Inverses: $a+(-a)=0$
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For group with $4$ elements
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|*|0|1|2|3|
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|---|---|---|---|---|
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|0|0|1|2|3|
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|1|1|2|3|0|
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|2|2|3|0|1|
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|3|3|0|1|2|
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#### Complex numbers
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Consider $\{1,i,-1,-i\}$ with multiplication.
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|*|1|i|-1|-i|
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|---|---|---|---|---|
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|1|1|i|-1|-i|
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|i|i|-1|-i|1|
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|-1|-1|-i|1|i|
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|-i|-i|1|i|-1|
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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 $\mathbb{Z}_4$.
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