Montreal Math Club

Mathematics students in Montreal - Étudiants de mathématiques à Montréal
 
HomeHome  CalendarCalendar  FAQFAQ  SearchSearch  MemberlistMemberlist  UsergroupsUsergroups  RegisterRegister  Log in  

Share
 

 [SOLVED] Group theory problem

Go down 
AuthorMessage
Bruno
Admin
Bruno

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

[SOLVED] Group theory problem Empty
PostSubject: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyWed Nov 11, 2009 5:46 pm



Last edited by Bruno on Fri Nov 13, 2009 7:53 pm; edited 1 time in total
Back to top Go down
View user profile http://mathclub.forumotion.com
Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyWed Nov 11, 2009 11:44 pm

The only automorphisms can be of the forms A(x)=X^m for some natural numbers m, now when is it an automorphism? Obviously when (m,n)=1, otherwise the order of elements don't agrees but an automorphism has to preserve the order bounce
Back to top Go down
View user profile
Bruno
Admin
Bruno

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyThu Nov 12, 2009 12:24 am

Mohammad wrote:
The only automorphisms can be of the forms A(x)=X^m for some natural numbers m

Why? Prove this directly.



Quote :
Obviously when (m,n)=1


Why? Give a proof.

The problem isn't trivial, I don't think you can get rid of it in two lines. pirat
Back to top Go down
View user profile http://mathclub.forumotion.com
Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyThu Nov 12, 2009 12:56 am

O.K, More details Exclamation
For n=p^{\alpha} ,where p is a prime number and \alpha is a natural number, the claim is obvious Smile
Now every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups, in our case we don't need "infinite cyclic groups" so... we are done bounce
Back to top Go down
View user profile
Bruno
Admin
Bruno

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyThu Nov 12, 2009 11:45 am

Good! cheers
It's obvious when n=p^alpha and p is an odd prime, because in that case the group of units of Z/nZ is cyclic. But when n=2^alpha?
Back to top Go down
View user profile http://mathclub.forumotion.com
Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyThu Nov 12, 2009 11:11 pm

Back to top Go down
View user profile
Bruno
Admin
Bruno

Posts : 184
Join date : 2009-09-15
Age : 32
Location : the infinite, frictionless plane of uniform density

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyFri Nov 13, 2009 8:08 pm

Awesome! Thanks bom
Back to top Go down
View user profile http://mathclub.forumotion.com
Mohammad
Descartes
Descartes
Mohammad

Posts : 100
Join date : 2009-11-05
Age : 35
Location : Right behind you

[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem EmptyFri Nov 13, 2009 8:31 pm

Basketball
Back to top Go down
View user profile
Sponsored content




[SOLVED] Group theory problem Empty
PostSubject: Re: [SOLVED] Group theory problem   [SOLVED] Group theory problem Empty

Back to top Go down
 
[SOLVED] Group theory problem
Back to top 
Page 1 of 1
 Similar topics
-
» Karmik-Theory
» Facebook Group
» First week discussion topic - group 2 - Emmy Awards/Tv Series
» Gassho From Texas
» PPMP to APP

Permissions in this forum:You cannot reply to topics in this forum
Montreal Math Club :: Mathematics :: Problem Solving :: Solved problems-
Jump to: