Let be a set, and let be a binary operation on . The structure is called a if the following hold.

1. For all one has .
2. There is a unique such that for all one has .
3. For all there is a unique such that .

We usually simply write when referring to the entire structure . The element from the second point is called the . The element from the third point is called the of and is usually denoted . One often simply writes in place of , and for every positive integer , is short for ( times).

• A binary operation on is simply a function from to (and its domain is all of ). As is customary, we write in place of .
• We usually think of as defining a multiplication on , 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 instead of . Similarly, when thinking of as addition we may use the symbol instead of .

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