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置换</font></b> </p>
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<p style="line-height: 200%"><a name="content-5-4-1-zhihuan"></a><b>置换</b><br>
设 X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">}是一个有穷集合。从集合
X 到 X 的<a href="content-5-2-1.htm#content-5-2-1-shuangshe">双射函数</a>，称为集合
X 中的置换或排列，并记作 P:X<img src="image/dao.GIF" width="15" height="9">X。集合
X 的基数 |X|=n 称为置换的阶。<br>
置换也可记为:</p>
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<p align="right" style="line-height: 200%">P=</td>
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<p style="line-height: 200%"><font size="6">(</font></p>
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<p style="line-height: 200%"> <img src="image/x1.GIF" width="11" height="11">
<img src="image/x2.GIF" width="12" height="11"> ... <img src="image/xn.GIF" width="12" height="11"></p>
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<p style="line-height: 200%"><font size="6">)</font></p>
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<p style="line-height: 200%">P(<img src="image/x1.GIF" width="11" height="11">)
P(<img src="image/x2.GIF" width="12" height="11">)... P(<img src="image/xn.GIF" width="12" height="11">)</p>
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<p style="line-height: 200%" align="center">或P={<<img src="image/x1.GIF" width="11" height="11">,P(<img src="image/x1.GIF" width="11" height="11">)>,<<img src="image/x2.GIF" width="12" height="11">,P(<img src="image/x2.GIF" width="12" height="11">)>,...,<<img src="image/xn.GIF" width="12" height="11">,P(<img src="image/xn.GIF" width="12" height="11">)>}</p>
<p style="line-height: 200%" align="left"> 显然,P(<img src="image/x1.GIF" width="11" height="11">)
P(<img src="image/x2.GIF" width="12" height="11">)... P(<img src="image/xn.GIF" width="12" height="11">)
<img border="0" src="image/shuyu.gif" width="13" height="11"> X</p>
<p style="line-height: 200%" align="left"> </p>
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<p style="line-height: 200%"><b>恒等置换</b> <br>
给定集合
X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">}。恒等函数
I<sub>X</sub>:X<img src="image/dao.GIF" width="15" height="9">X
是集合 X 中的恒等变换，并记作 P<sub>X</sub>，</p>
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<p align="right" style="line-height: 200%">P<sub>X</sub>=</td>
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<p style="line-height: 200%"><font size="6">(</font></p>
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<td width="103%" align="center">
<p style="line-height: 200%"> <img src="image/x1.GIF" width="11" height="11">
<img src="image/x2.GIF" width="12" height="11"> ... <img src="image/xn.GIF" width="12" height="11"></p>
</td>
<td width="11%" rowspan="2">
<p style="line-height: 200%"><font size="6">)</font></p>
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<p style="line-height: 200%"> <img src="image/x1.GIF" width="11" height="11">
<img src="image/x2.GIF" width="12" height="11"> ... <img src="image/xn.GIF" width="12" height="11"></p>
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<p style="line-height: 200%"><b>反置换</b> <br>
给定集合X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">}。X
中的任意 n 阶置换 </p>
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<p align="right" style="line-height: 200%">P=</td>
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<p style="line-height: 200%"><font size="6">(</font></p>
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<td width="103%" align="center">
<p style="line-height: 200%"> <img src="image/x1.GIF" width="11" height="11">
<img src="image/x2.GIF" width="12" height="11"> ... <img src="image/xn.GIF" width="12" height="11"></p>
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<p style="line-height: 200%"><font size="6">)</font></p>
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<p style="line-height: 200%">P(<img src="image/x1.GIF" width="11" height="11">)
P(<img src="image/x2.GIF" width="12" height="11">)... P(<img src="image/xn.GIF" width="12" height="11">)</p>
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<p style="line-height: 200%">的反置换是</p>
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<p align="right" style="line-height: 200%">P<sup>-1</sup>=</td>
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<p style="line-height: 200%"><font size="6">(</font></p>
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<p style="line-height: 200%"> P(<img src="image/x1.GIF" width="11" height="11">)
P(<img src="image/x2.GIF" width="12" height="11">)... P(<img src="image/xn.GIF" width="12" height="11">)</p>
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<td width="10%" rowspan="2">
<p style="line-height: 200%"><font size="6">)</font></p>
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<td width="104%" align="center">
<p style="line-height: 200%"> <img src="image/x1.GIF" width="11" height="11">
<img src="image/x2.GIF" width="12" height="11"> ... <img src="image/xn.GIF" width="12" height="11"></p>
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</center>
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<p style="line-height: 200%"> </p>
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<p style="line-height: 200%"><b>置换的循环</b> </li>
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<p style="line-height: 200%"> 置换的合成可记为:,P<sup>2</sup>,P<sup>3</sup>,……</p>
<p style="line-height: 200%"> 对于P:X<img src="image/dao.GIF" width="15" height="9">X这个n阶置换,其中X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">},若对
<img border="0" src="image/xi.GIF" width="10" height="11"> <img border="0" src="image/shuyu.gif" width="13" height="11">
X 考虑一下序列:<br>
P<sup>0</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">)
P(<img border="0" src="image/xi.GIF" width="10" height="11">) P<sup>2</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">)
P<sup>3</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">) … <img border="0" src="image/shuyu.gif" width="13" height="11">
X <br>
因为X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">}为有限集合,所以序列中必然出现重复的项,即:存在
k,t 满足0<img border="0" src="image/xiaodeng.GIF" width="9" height="10">k<img border="0" src="image/xiaodeng.GIF" width="9" height="10">t<img border="0" src="image/xiaodeng.GIF" width="9" height="10">n,
使得 P<sup>k</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">)=P<sup>t</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">).</p>
<p style="line-height: 200%"> 称P<sup>k</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">)
P<sup>k+1</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">) … P<sup>t-1</sup>(<img border="0" src="image/xi.GIF" width="10" height="11">)
为置换中 <img border="0" src="image/xi.GIF" width="10" height="11"> 的一个循环.</p>
<p style="line-height: 200%"> 令r=t-k, 称 r
为置换P中 <img border="0" src="image/xi.GIF" width="10" height="11"> 的循环的阶.</p>
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在一个置换中,每个元素所在的循环不相交,在同一个循环中的元素的循环相等.</p>
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类似于等价关系的等价类,我们可以按照置换的循环把X={<img src="image/x1.GIF" width="11" height="11">,<img src="image/x2.GIF" width="12" height="11">,...,<img src="image/xn.GIF" width="12" height="11">}分成若干个互不相交的部分.</p>
<p style="line-height: 200%"> 如:</p>
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<p align="right" style="line-height: 200%">P=</td>
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<p style="line-height: 200%"><font size="6">(</font></p>
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<p style="line-height: 200%"> a b c
d e f g h </p>
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<p style="line-height: 200%"><font size="6">)</font></p>
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<p style="line-height: 200%">h g b
f e d a c</td>
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<p style="line-height: 200%"> 其中含有 a 的循环有:<input type="text" name="T1" size="27"><br>
含有 d 的循环有:<input type="text" name="T1" size="27"><br>
含有 e 的循环有:<input type="text" name="T1" size="27"></p>
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</p>
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<p style="line-height: 200%"> <img src="IMAGE/yuancheng_2.gif" width="100" height="69"></div>
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<p style="line-height: 200%"> </p>
<p style="line-height: 150%"> </p><p align="right"><b><a href="contentFrame-mulu.htm"><<back</a></b>
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