# Compute Right Coset of (1 2) in S3 X Z2

• FanofAFan
In summary, the question is asking for the coset of ((1 2), [1]) in S3xZ2, which is just a subset of S3xZ2.
FanofAFan
Compute the right coset of ((1 2), [1])) in S3 X Z2.

Do I use the given as H (the subgroup) and find right cosets for just (1 2) and then coset for [1] separately. Basically I'm just confused on what to do.

FanofAFan said:
Compute the right coset of ((1 2), [1])) in S3 X Z2.

Do I use the given as H (the subgroup) and find right cosets for just (1 2) and then coset for [1] separately. Basically I'm just confused on what to do.

The right coset would be the set of all elements {$$\alpha$$} such that $$\alpha$$=((1 2),[1])(p,n) where p is an element of S3 and n is an element of Z2. I would list the elements of S3 X Z2 (which would be like ((1 2 3),[2]),((1 2 3),[2])...- there should be 12 elements I believe?), and operate ((1 2), [1]) with each of them. Then those elements are the elements of the right coset of ((1 2),[1]).

Right? I'm just a group theory student myself... but that's how I would go about that.

The question doesn't make sense to me. Isn't ((1 2), [1]) just an element of S3xZ2, not a subgroup? Is the question referring to the subgroup generated by that element?

The question is referring to ((12), [1]) being the subgroup... I don't really know how to explain it further... hence the confusion.

@sephy, I worked it out the way you explain it

Last edited:
It's no surprise you're confused then. You should ask your instructor for clarification.

vela said:
The question doesn't make sense to me. Isn't ((1 2), [1]) just an element of S3xZ2, not a subgroup? Is the question referring to the subgroup generated by that element?

The element is its own inverse, so the subgroup generated by that element would just be itself and the identity element. Maybe you are meant to assume that?

Either that or the word "coset" is being mis-used- vela's right, it's not a coset unless it's made with a subgroup. Without the identity ((1 2), [1]) is just a subset of S3xZ2. You could still compute the elements of Ha, but Ha would not technically be considered a coset.

## 1. What is the definition of a "right coset"?

A right coset is a subgroup of a larger group that is obtained by multiplying all elements of the larger group by a specific element on the right side. In this case, we are finding the right coset of the permutation (1 2) in the direct product of the symmetric group S3 and the cyclic group Z2.

## 2. How is the right coset of (1 2) in S3 X Z2 computed?

To compute the right coset of (1 2), we need to multiply (1 2) with all elements of the direct product S3 X Z2. This will give us a subgroup of S3 X Z2 that contains all elements that can be obtained by multiplying (1 2) with an element on the right side.

## 3. What is the significance of finding the right coset of (1 2) in S3 X Z2?

Finding the right coset of (1 2) in S3 X Z2 helps us understand the structure of the larger group. It allows us to identify a subgroup that is related to (1 2) and to see how the elements of this subgroup are related to each other under the operation of multiplication.

## 4. How many elements are in the right coset of (1 2) in S3 X Z2?

The number of elements in the right coset of (1 2) in S3 X Z2 depends on the size of the larger group and the specific element (1 2). Since S3 has 6 elements and Z2 has 2 elements, the direct product S3 X Z2 has 6*2=12 elements. Therefore, the right coset of (1 2) in S3 X Z2 will also have 12 elements.

## 5. Can the right coset of (1 2) in S3 X Z2 be written in a simpler form?

Yes, the right coset of (1 2) in S3 X Z2 can be written in a simpler form using the notation of sets. If we let H be the subgroup generated by (1 2) in S3 X Z2, then the right coset of (1 2) in S3 X Z2 can be written as {(1 2)x | x∈H}. This notation represents the set of all elements that can be obtained by multiplying (1 2) with an element from H on the right side.

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