From 7400bd41a43c2f8a33ce639f15724bffa02d5360 Mon Sep 17 00:00:00 2001 From: Brandon Rozek Date: Sat, 11 Jun 2022 20:09:26 -0400 Subject: [PATCH] Changed PDF notes to markdown notes --- content/notes/_index.md | 1 + content/notes/groups-abstract-algebra.md | 46 +++++++++++++++++++++++ static/files/groupssimplified.pdf | Bin 104846 -> 0 bytes 3 files changed, 47 insertions(+) create mode 100644 content/notes/groups-abstract-algebra.md delete mode 100644 static/files/groupssimplified.pdf diff --git a/content/notes/_index.md b/content/notes/_index.md index e6a600a..f7e629b 100644 --- a/content/notes/_index.md +++ b/content/notes/_index.md @@ -15,6 +15,7 @@ purposes. ## Courses - [Probability and Statistical Inference I](stat381) - [Real Analysis Quick Facts](realanalysis) +- [Groups in Abstract Algebra](groups-abstract-algebra) - [Some Abstract 2 Definitions](abstract2def) - [Handy Facts about Quadratic Congruences](quadraticcongruences) - [Bayesian Statistics](bayesianstatistics) diff --git a/content/notes/groups-abstract-algebra.md b/content/notes/groups-abstract-algebra.md new file mode 100644 index 0000000..59c74b1 --- /dev/null +++ b/content/notes/groups-abstract-algebra.md @@ -0,0 +1,46 @@ +--- +title: "Groups in Abstract Algebra" +draft: false +tags: [] +math: true +--- + +Let us have a set $G$ together with some binary operation $*$. +We will use multipicative notation where $ab = a * b$. +Let $x, y, z \in G$. If $\langle G, *\rangle$ has the +following properties: +1. $(xy)z = x(yz)$ +2. $ex = x$ +3. $x^{-1}x = e$ + +for some fixed $e \in G$, then we say that $\langle G, *\rangle$ is a group. +In my class, we were also told to show that $xe = x$ and $xx^{-1} = e$. +However, these can be derived by the prior three properties. + +## Prove $xx^{-1} = e$ + +$$ +\begin{align*} +e &= (xx^{-1})^{-1}(xx^{-1}) \\\\ + &= (xx^{-1})^{-1}(x(ex^{-1})) \\\\ + &= (xx^{-1})^{-1}(x((x^{-1}x)x^{-1})) \\\\ + &= (xx^{-1})^{-1}(x(x^{-1}x)x^{-1}) \\\\ + &= (xx^{-1})^{-1}((xx^{-1})(xx^{-1})) \\\\ + &= ((xx^{-1})^{-1}(xx^{-1}))(xx^{-1}) \\\\ + &= e(xx^{-1}) \\\\ + &= xx^{-1} \\\\ +\end{align*} +$$ + +## Prove $xe = x$ + +We can use the last proof to solve this faster. + +$$ +\begin{align*} +x &= ex \\\\ + &= (xx^{-1})x \\\\ + &= x(x^{-1}x) \\\\ + &= xe +\end{align*} +$$ diff --git a/static/files/groupssimplified.pdf b/static/files/groupssimplified.pdf deleted file mode 100644 index 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