mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-11-22 16:26:28 -05:00
47 lines
1 KiB
Markdown
47 lines
1 KiB
Markdown
|
---
|
||
|
|
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.
|
||
|
|
|
||
|
|
$$
|
||
|
|
\begin{align*}
|
||
|
|
x &= ex \\\\
|
||
|
|
&= (xx^{-1})x \\\\
|
||
|
|
&= x(x^{-1}x) \\\\
|
||
|
|
&= xe
|
||
|
|
\end{align*}
|
||
|
|
$$
|