Finding Surjective Homomorphisms from Symmetric Groups to Cyclic Groups

[Indran]

Hello,
I am having difficulty with the following problem in Group theory:

How do you positive integers r such that there is a surjective homomorphism from S_n (This is the symmetric group of order n) to
C_r (This is the cyclic group order r) for some n ?
I am not sure where to even start and any pointers in right direction will
be greatly appreciated.

 

[AKG]

What happens when r is greater than n?

 

[Indran]

Groups

What happens when r is greater than n?

Well, on the face of it. since r>n I would have said that therecannot be a surjective relationship, but if say, S_3 ={1 2 3} and C_4 (thus r>n) is
{e, a, a^2, a^3} the order of S_3 is 3! = 6 and the order of C_4 is 4
so there could be a surjective homomorphism. I am stuck at this point as I need to find values of r that would give this homomorphism for some n.
Can you help?

 

[matt grime]

You're ignoring many importan aspects of the structure of groups and homomorphisms. Do you know the isomorphism theorems? Your problem asks you when is there a normal subgroup H of S_n with S_n/H isomorphic to C_r. There can't be a surjective hom from S_3 to C_4 because 6/4 is not an integer. Homs alse have certain properties to do with orders of elements as well. It's impossible to send any element of S_n to any element of order 4 in any other group at all since 4 does not divide the order of any element of S_4 either.

(I presume the question did not ask you to find *all* n and r with this property - it is possible, but would require you to know too much at this stage).

 

[Indran]

Groups

You're ignoring many importan aspects of the structure of groups and homomorphisms. Do you know the isomorphism theorems? You're problem asks you when is there a normal subgroup H of S_n with S_n/H isomorphic to C_r. There can't be a surjective hom from S_3 to C_4 because 6/4 is not an integer. Homs alse have certain properties to do with orders of elements as well. It's impossible to send any element of S_n to any element of order 4 in any other group at all since 4 does not divide the order of any element of S_4 either.

(I presume the question did not ask you to find *all* n and r with this property - it is possible, but would require you to know too much at this stage).

Thanks for the clear reply.
Yes, I looked up the Isomorphism theorem and now can understand what you are saying. The question does ask for *all* r for some n that will give a homomorphism from S_n to C_r. How can this be done?

 

[AKG]

{e, a, a^2, a^3} the order of S_3 is 3! = 6 and the order of C_4 is 4 so there could be a surjective homomorphism. I am stuck at this point as I need to find values of r that would give this homomorphism for some n.

Think harder.