website/content/notes/groups-abstract-algebra.md

title	draft	tags	math
Groups in Abstract Algebra	false		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*}