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#+title: group
#+author: Preston Pan
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* Definition
A group is an [[id:1b1b522e-d4de-4832-9ca4-c6d1cfee27e6][ordered pair]] \((G, *)\) where \(G\) is a set and \(*\) is a binary operation (operation defined between two members of set G) defined such that:
\begin{align*}
a * b \in G \\
\exists e : a * e = a
\end{align*}
where the operation \(*\) is said to be closed under \(G\), and \(e\) is called the /identity/ of group \((G, *)\).
** Associativity
This is the property such that:
\begin{align*}
(a * b) * c = a * (b * c)
\end{align*}
** inverse
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An inverse is defined as follows:
\begin{align*}
\forall a \exists a^{-1} : a * a^{-1} = e
\end{align*}

* Motivation
In [[id:ece8bf94-4e3c-4939-a77a-9949c1ec0dc6][physics]], natural phenomena including conservation laws follow from group symmetries.