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definition:group
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definition:group [2013/08/21 09:27] joshuawiscons |
definition:group [2013/08/21 13:01] (current) bmwoodruff [External links] |
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| - $\textbf{[Identity]}$ There is a unique $e\in G$ such that for all $x\in G$ one has $x * e = e* x = x$. | - $\textbf{[Identity]}$ 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 $x* y = y* x = e$. | - $\textbf{[Inverses]}$ 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). | 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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| - | {{tag> | + | {{tag> |
definition/group.1377091667.txt.gz · Last modified: 2013/08/21 09:27 by joshuawiscons
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