# Prove Subset can't be coset with two different subgroups

• FanofAFan
In summary, a subset is a smaller group of elements that are all part of a larger set, while a coset is a subset of a group obtained by multiplying all elements of the group by a fixed element. A subset cannot be a coset with two different subgroups because a coset is defined by the multiplication of a fixed element and all elements of a group, while a subset does not have this same definition. This can be proven by using the definition of a coset and the properties of subgroups.
FanofAFan
Prove that a subset S of a group G cannot be a right coset of two different subgroups of G.

I having a hard time proving this... but this is what I have so far;
Fix z in Hz and since z = ez where e is in H. Then assume z in Hy by the right coset equation then with x = z we see that Hz = Hy and if Hx intersection of Hy $$\neq$$ to the empty set... that as far as i got. Please help

## 1. What is a subset?

A subset is a set that contains elements from another set. It is a smaller group of elements that are all part of a larger set.

## 2. What is a coset?

A coset is a subset of a group that is obtained by multiplying all elements of the group by a fixed element. It can also be defined as the set of all products of a fixed element and the elements of the group.

## 3. Can a subset be a coset with two different subgroups?

No, a subset cannot be a coset with two different subgroups. This is because a coset is defined by the multiplication of a fixed element and all elements of a group, while a subset is simply a smaller group of elements from a larger set. Therefore, a subset cannot have the same definition as a coset.

## 4. Why can't a subset be a coset with two different subgroups?

A subset cannot be a coset with two different subgroups because it would violate the definition of a coset. A coset is defined by the multiplication of a fixed element and all elements of a group, and a subset does not have this same definition. Therefore, a subset cannot be a coset with two different subgroups.

## 5. How can we prove that a subset cannot be a coset with two different subgroups?

We can prove that a subset cannot be a coset with two different subgroups by using the definition of a coset and the properties of subgroups. We can show that a subset does not have the same definition as a coset, and therefore cannot be a coset with two different subgroups.

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