Proving something about right cosets of distinct subgroups of a group

[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.

 

[Dick]

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?