S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Wed, 26 Nov 2014 21:06:31 UTC

All times are UTC [ DST ]




Post new topic Reply to topic  [ 4 posts ] 
Author Message
 Post subject: Group Theory
PostPosted: Fri, 22 Oct 2010 00:35:42 UTC 
Offline
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 14246
Location: Austin, TX
Show that there is a transitive action of S_5 on a set with six elements.

Hint:
Spoiler:
Sylow's Theorem

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination


Top
 Profile  
 
 Post subject: Re: Group Theory
PostPosted: Fri, 22 Oct 2010 09:32:58 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6990
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
Shadow wrote:
Show that there is a transitive action of S_5 on a set with six elements.

Hint:
Spoiler:
Sylow's Theorem


Alternative hint:
Spoiler:
Character table of S_6

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
 Post subject: Re: Group Theory
PostPosted: Tue, 19 Jul 2011 19:00:47 UTC 
Offline
S.O.S. Newbie

Joined: Tue, 19 Jul 2011 17:58:35 UTC
Posts: 1
Shadow wrote:
Show that there is a transitive action of S_5 on a set with six elements.


By S_5 you refer to the Symmetric Group, that is, all bijections operating on 5 symbols? If so, how is S_5 supposed to act on a set of 6? (Please forgive my ignorance -- I am a beginner in group theory)


Top
 Profile  
 
 Post subject: Re: Group Theory
PostPosted: Tue, 19 Jul 2011 21:40:27 UTC 
Offline
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 14246
Location: Austin, TX
Ralph Dratman wrote:
Shadow wrote:
Show that there is a transitive action of S_5 on a set with six elements.


By S_5 you refer to the Symmetric Group, that is, all bijections operating on 5 symbols? If so, how is S_5 supposed to act on a set of 6? (Please forgive my ignorance -- I am a beginner in group theory)


Yes, that is exactly what I mean.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 4 posts ] 

All times are UTC [ DST ]


Who is online

Users browsing this forum: No registered users


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA