====== Group ======

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

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==== Remarks ==== 
  * A [[wp>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 $e$.


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==== $\LaTeX$ version ====
<file tex definition.group.tex>
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% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} 
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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}
</file>

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==== External links ====
  * [[wp>Group (mathematics)|Group]]


{{tag>definition needsreview rjosh rben}}