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athyra

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## 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.