# Proving something about right cosets of distinct subgroups of a group

• athyra
In summary, the problem is to prove that a subset S of a group G cannot be a right coset of two different subgroups of G. The relevant equations are those involving the definitions of right cosets. The attempt at a solution involves assuming that a subset F of G is equal to right cosets of two distinct subgroups of G. The idea is to show that if m is in one subgroup, it must also be in the other, leading to a contradiction. However, this approach does not take into account the fact that the identity must be contained in every right coset. This means that if F is equal to two right cosets, one of them must be the entire group G, making it impossible for F to be
athyra

## Homework Statement

Prove that a subset S of a group G cannot be a right coset of two different subgroups of G.

## Homework Equations

The relevant equations are those involving the definitions of right cosets.
a is in the right coset of subgroup H of group G if a = hg where h is in H and g is in G, possibly in H.

## The Attempt at a Solution

First I let a subset F of G be equal to right cosets of two distinct subgroups of G. So let H and K be subgroups of G such that H doesn't equal K. Now assume F = Hg_1 = Kg_2, where g_1, and g_2 are both in G. So F is now equal to two right cosets of distinct subgroups of G. So my idea was to let m be in H, and show it must be in K. I believe the argument for this will be reversible so it will be almost identical showing that if m is in K it must be in H. Then I would have found my contradiction. So what to do once assuming m is in H is where I am stuck. Any help would be greatly appreciated, as I have never written in a forum for help before.

Showing if m is in H then it's in K and vice-versa would do it alright. But that's what you want to show in general to show two sets are equal and doesn't have anything in particular to do with groups or cosets. So it's not much of a start. Try thinking about this. If F=Hg_1=Kg_2 then H=K(g_2)(g_1)^(-1). That means H is a right coset of K, true? H contains the identity. How many right cosets of K contain the identity?

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## 1. What is a right coset?

A right coset is a subset of a group that is obtained by multiplying all the elements of the subgroup by a fixed element in the group from the right.

## 2. What is the significance of proving something about right cosets of distinct subgroups?

Proving something about right cosets of distinct subgroups can help us understand the structure and properties of a group, as well as its subgroups.

## 3. How do you prove something about right cosets of distinct subgroups?

To prove something about right cosets of distinct subgroups, you can use various techniques such as the Lagrange's theorem, the coset decomposition theorem, or direct proof using the definition of right cosets.

## 4. Can a right coset be equal to the entire group?

Yes, a right coset can be equal to the entire group if the subgroup used to form the coset is the identity element of the group.

## 5. How do right cosets relate to normal subgroups?

A normal subgroup is a subgroup that is closed under conjugation. This means that the left and right cosets of a normal subgroup are the same, making it easier to work with them in proofs and calculations.

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