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10-1100/Homework Assignment 2 - Revision history
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Drorbn at 17:29, 20 October 2010
2010-10-20T17:29:00Z
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>## What it the least integer <math>n</math> for which the symmetric group <math>S_n</math> contains an element of order 18?</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>## What it the least integer <math>n</math> for which the symmetric group <math>S_n</math> contains an element of order 18?</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>## What is the maximal order of an element in <math>S_{<del style="font-weight: bold; text-decoration: none;">52</del>}</math>? (That is, of a shuffling of a deck of cards?)</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>## What is the maximal order of an element in <math>S_{<ins style="font-weight: bold; text-decoration: none;">26</ins>}</math>? (That is, of a shuffling of<ins style="font-weight: bold; text-decoration: none;"> the red cards within</ins> a deck of cards?)</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># (Selick) Let <math>H</math> be a subgroup of index 2 in a group <math>G</math>. Show that <math>H</math> is normal in <math>G</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># (Selick) Let <math>H</math> be a subgroup of index 2 in a group <math>G</math>. Show that <math>H</math> is normal in <math>G</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Let <math>\sigma\in S_{20}</math> be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer <math>C_{S_{20}}(\sigma)</math> of <math>\sigma</math>?</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># Let <math>\sigma\in S_{20}</math> be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer <math>C_{S_{20}}(\sigma)</math> of <math>\sigma</math>?</div></td>
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Drorbn
https://drorbn.net/index.php?title=10-1100/Homework_Assignment_2&diff=9608&oldid=prev
Drorbn at 01:23, 13 October 2010
2010-10-13T01:23:16Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:23, 12 October 2010</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This assignment is due at class time on Thursday, October 21, 2010.</div></td>
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Drorbn
https://drorbn.net/index.php?title=10-1100/Homework_Assignment_2&diff=9606&oldid=prev
Drorbn at 01:17, 13 October 2010
2010-10-13T01:17:41Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:17, 12 October 2010</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Solve the following questions===</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># (Selick)</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>## What it the least integer <math>n</math> for which the symmetric group <math>S_n</math> contains an element of order 18?</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>## What is the maximal order of an element in <math>S_{52}</math>? (That is, of a shuffling of a deck of cards?)</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># (Selick) Let <math>H</math> be a subgroup of index 2 in a group <math>G</math>. Show that <math>H</math> is normal in <math>G</math>.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># Let <math>\sigma\in S_{20}</math> be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer <math>C_{S_{20}}(\sigma)</math> of <math>\sigma</math>?</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div># (Selick) Let <math>G</math> be a group of odd order. Show that <math>x</math> is not conjugate to <math>x^{-1}</math> unless <math>x=e</math>.</div></td>
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<td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>## Let <math>G</math> be a group and let <math>H</math> be a subgroup of finite index. Prove that there is a normal subgroup <math>N</math> of <math>G</math>, contained in <math>H</math>, so that <math>(G:N)</math> is also finite. (Hint: Let <math>(G:H)=n</math> and find a morphism <math>G\to S_n</math> whose kernel is contained in <math>H</math>.)</div></td>
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Drorbn
https://drorbn.net/index.php?title=10-1100/Homework_Assignment_2&diff=9541&oldid=prev
Drorbn at 18:28, 11 October 2010
2010-10-11T18:28:56Z
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<p><b>New page</b></p><div>{{10-1100/Navigation}}<br />
{{In Preparation}}<br />
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This assignment is due at class time on Thursday, October 21, 2010.<br />
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===Solve the following questions===</div>
Drorbn