Let $G$ be a set, and let $*$ be a binary operation on $G$. The structure $\mathbb{G} = (G,*)$ is called a $\textdef{group}$ if the following hold.

1. $\textbf{[Associativity]}$ For all $x,y,z\in G$ one has $(x* y)* z = x* (y* z)$.
2. $\textbf{[Identity]}$ There is a unique $e\in G$ such that for all $x\in G$ one has $x * e = e* x = x$.
3. $\textbf{[Inverses]}$ For all $x\in G$ there is a unique $y\in G$ such that $x* y = y* x = e$.

\end{enumerate} We usually simply write $G$ when referring to the entire structure $\mathbb{G}$. The element $e$ from the second point is called the $\textdef{identity}$. The element $y$ from the third point is called the $\textdef{inverse}$ of $x$ and is usually denoted $x^{-1}$. One often simply writes $xy$ in place of $x* y$, and for every positive integer $n$, $x^n$ is short for $x* x* \cdots * x$ ($n$ times).

• A binary operation on $G$ is simply a function from $G\times G$ to $G$ (and its domain is all of $G\times G$). As is customary, we write $x*y$ in place of $*(x,y)$.
• We usually think of $*$ as defining a multiplication on $G$, though often this is better interpreted as addition. In the later case, we may use the symbol $+$ instead of $*$.
• When thinking of $*$ as multiplication we often use the symbol $1$ instead of $e$. Similarly, when thinking of $*$ as addition we may use the symbol $0$ instead of $e$.

definition.group.tex

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\begin{definition}
Let $G$ be a set, and let $*$ be a binary operation on $G$. The structure $\mathbb{G} = (G,*)$ is called a \textdef{group} if the following hold.
\begin{enumerate}
\item \textbf{[Associativity]} For all $x,y,z\in G$ one has $(x* y)* z = x* (y* z)$.
\item \textbf{[Identity]} There is a unique $e\in G$ such that for all $x\in G$ one has $x * e = e* x = x$. 
\item \textbf{[Inverses]} For all $x\in G$ there is a unique $y\in G$ such that $x* y = y* x = e$.
\end{enumerate}
We usually simply write $G$ when referring to the entire structure $\mathbb{G}$. The element $e$ from the second point is called the \textdef{identity}. The element $y$ from the third point is called the \textdef{inverse} of $x$ and is usually denoted $x^{-1}$. One often simply writes $xy$ in place of $x* y$, and for every positive integer $n$, $x^n$ is short for $x* x* \cdots * x$ ($n$ times).
\end{definition}

• Group

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