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--- definition:group [2013/08/08 02:11]
bmwoodruff
+++ definition:group [2013/08/21 13:01] (current)
bmwoodruff [External links]
@@ Line 1: / Line 1: @@
-(Please feel free to adjust this, or give a different defintion.  I'm hoping to let the students come up with a definition of a group on their own, after playing with several different scenarios.
+====== Group ======
-A group is a set $G$ together with a binary operation $\cdot$ that is
+====Definition====
-# associative, meaning $a\cdot (b\cdot c) = (a\cdot b)\cdot c$,
+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.
-# ... be back later.
+  - $\textbf{[Associativity]}$ 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).
+----
+==== 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 $e$.
+----
+==== $\LaTeX$ version ====
+<file tex definition.group.tex>
+%%%%%
+% DEPENDENCIES
+% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}}
+%%%%%
+\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>
+----
+==== External links ====
+  * [[wp>Group (mathematics)|Group]]
+{{tag>definition needsreview rjosh rben}}

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