Montreal Math ClubMathematics students in Montreal - Étudiants de mathématiques à Montréal

 Montreal Math Club :: Mathematics :: Problem Solving :: Solved problems Share

# [SOLVED] Group theory problem

AuthorMessage

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

 Subject: [SOLVED] Group theory problem   Wed Nov 11, 2009 5:46 pm Last edited by Bruno on Fri Nov 13, 2009 7:53 pm; edited 1 time in total

Descartes

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

 Subject: Re: [SOLVED] Group theory problem   Wed 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

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

 Subject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 12:24 am Mohammad wrote:The only automorphisms can be of the forms A(x)=X^m for some natural numbers mWhy? Prove this directly.Quote :Obviously when (m,n)=1Why? Give a proof.The problem isn't trivial, I don't think you can get rid of it in two lines.

Descartes

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

 Subject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 12:56 am O.K, More details For n=p^{\alpha} ,where p is a prime number and \alpha is a natural number, the claim is obvious 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

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

 Subject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 11:45 am Good! 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?

Descartes

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

 Subject: Re: [SOLVED] Group theory problem   Thu Nov 12, 2009 11:11 pm

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

 Subject: Re: [SOLVED] Group theory problem   Fri Nov 13, 2009 8:08 pm Awesome! Thanks

Descartes

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

 Subject: Re: [SOLVED] Group theory problem   Fri Nov 13, 2009 8:31 pm

 Subject: Re: [SOLVED] Group theory problem

 [SOLVED] Group theory problem
 Page 1 of 1
 Similar topics
» Karmik-Theory