# Differences

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groups [2010/07/29 18:30] 127.0.0.1 external edit |
groups [2012/06/15 23:11] (current) jipsen |
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It follows that $e$ is a right-identity and that $^{-1}$gives a right | It follows that $e$ is a right-identity and that $^{-1}$gives a right | ||

inverse: $xe=x$, $xx^{-1}=e$. | inverse: $xe=x$, $xx^{-1}=e$. | ||

- | |||

- | This definition shows that groups form a variety. | ||

==Morphisms== | ==Morphisms== | ||

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permutations of a sets $X$, with composition, inverse, and identity map. | permutations of a sets $X$, with composition, inverse, and identity map. | ||

- | Example 1: The general linear group $\langle GL_{n}(V),\cdot | + | Example 2: The general linear group $\langle GL_{n}(V),\cdot |

,^{-1},I_{n}\rangle $, the collection of invertible $n\times n$ | ,^{-1},I_{n}\rangle $, the collection of invertible $n\times n$ | ||

matrices over a vector space $V$, with matrix multiplication, inverse, and | matrices over a vector space $V$, with matrix multiplication, inverse, and | ||

identity matrix. | identity matrix. | ||

- | |||

- | Example 1: [[Groups (mace)]] | ||

- | |||

====Basic results==== | ====Basic results==== | ||

- | [[Groups (otter)]] | ||

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f(18)= &5\\ | f(18)= &5\\ | ||

\end{array}$ | \end{array}$ | ||

+ | |||

Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html | Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html | ||

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