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--- definition:group [2013/08/15 11:11]
joshuawiscons created
+++ definition:group [2013/08/21 13:01] (current)
bmwoodruff [External links]
@@ Line 2: / Line 2: @@
 ====Definition====
-Let $G$ be a [[wp>Set (mathematics)|Set]] together with a function $m:G\times G \rightarrow G$, a function $i:G \rightarrow G$, and a distinguished element $e\in G$. The structure $\mathbb{G} = (G,m,i,e)$ is called a $\textdef{group}$ if the following hold for all $x,y,z \in G$; we write $xy$ in place of $m(x,y)$ and $x^{-1}$ in place of $i(x)$.
+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.
-  - $(xy)z = x(yz)$
+  - $\textbf{[Associativity]}$ For all $x,y,z\in G$ one has $(x* y)* z = x* (y* z)$.
-  - $xx^{-1} = x^{-1}x = e$
+  - $\textbf{[Identity]}$ There is a unique $e\in G$ such that for all $x\in G$ one has $x * e = e* x = x$.
-  - $xe = ex = x$
+  - $\textbf{[Inverses]}$ 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 $\mathbb{G}$.
+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 ====
-  * We usually think of $m$ as defining a multiplication on $G$, though often this is better interpreted as addition.
+  * 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 $i$ as defining inversion on $G$, though often this is better interpreted as negation.
+  * 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 $m$ as multiplication we often use the symbol $1$ instead of $e$. Similarly, When thinking of $m$ as addition we may use the symbol $0$ instead of $e$.
+  * 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 ====
-<code>
+<file tex definition.group.tex>
 %%%%%
 % DEPENDENCIES
@@ Line 23: / Line 23: @@
 %%%%%
 \begin{definition}
-Let $G$ be a set together with a function $m:G\times G \rightarrow G$, a function $i:G \rightarrow G$, and a distinguished element $e\in G$. The structure $\mathbb{G} = (G,m,i,e)$ is called a \textdef{group} if the following hold for all $x,y,z \in G$; we write $xy$ in place of $m(x,y)$ and $x^{-1}$ in place of $i(x)$.
+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 $(xy)z = x(yz)$
+\item \textbf{[Associativity]} For all $x,y,z\in G$ one has $(x* y)* z = x* (y* z)$.
-\item $xx^{-1} = x^{-1}x = e$
+\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 $xe = ex = x$
+\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 $\mathbb{G}$.
+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}
-</code>
+</file>
 ----
@@ Line 38: / Line 38: @@
-{{tag>definition}}
+{{tag>definition needsreview rjosh rben}}

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