# Finding Surjective Homomorphisms from Symmetric Groups to Cyclic Groups

• Indran
In summary, the conversation discusses the difficulty of finding surjective homomorphisms between symmetric groups and cyclic groups, specifically when the order of the cyclic group is larger than the order of the symmetric group. The person asking for help has already looked into the isomorphism theorems but is unsure how to find all values of r that would give a homomorphism for some n. The other person suggests thinking harder about the properties of groups and homomorphisms.
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.

What happens when r is greater than n?

Groups

AKG said:
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?

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).

Last edited:
Groups

matt grime said:
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).

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?

Indran said:
{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.

## 1. What is a problem in Group theory?

A problem in Group theory refers to a mathematical question or puzzle related to the study of groups, which are algebraic structures that consist of a set of elements and an operation. These problems often involve finding patterns, symmetries, or relationships between elements within a group.

## 2. What are some common examples of problems in Group theory?

Some common problems in Group theory include the classification of finite simple groups, the existence of certain types of subgroups within a group, and the determination of the order (or number of elements) of a group.

## 3. How do mathematicians approach solving problems in Group theory?

Mathematicians use various techniques and strategies to solve problems in Group theory, such as studying specific examples, using mathematical proofs, and applying concepts from other areas of mathematics, such as number theory and topology.

## 4. What are the applications of Group theory in real-world problems?

Group theory has many applications in real-world problems, such as in cryptography, chemistry, and physics. For example, group theory is used in cryptography to create secure codes and in chemistry to understand the properties of molecules. In physics, group theory is used to describe symmetries in physical systems.

## 5. Are there any open problems in Group theory?

Yes, there are still many open problems in Group theory that have yet to be solved. Some notable examples include the inverse Galois problem, which asks whether every finite group can be the Galois group of a polynomial, and the Burnside problem, which asks whether a finitely generated group with bounded exponent must be finite.

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