# Subgroup containment implies coset containment

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## Statement

### Verbal statement

**For left cosets**: If one subgroup of a group is contained in another, then every left coset of the subgroup is contained in a left coset of the other subgroup.

**For right cosets**: If one subgroup of a group is contained in another, then every right coset of the subgroup is contained in a right coset of the other subgroup.

### Statement with symbols

**For left cosets**: Suppose are subgroups. Then, every Left coset (?) of is contained in exactly one left coset of .

**For right cosets**: Suppose are subgroups. Then, every Right coset (?) of is contained in exactly one right coset of .

## Related facts

- Coset containment implies subgroup containment: If a left coset of one subgroup is contained in a left coset of another, then the subgroup containment also holds.
- Nonempty intersection of cosets is coset of intersection: If the intersection of a collection of left cosets of subgroups is nonempty, it is a coset of the intersection of the corresponding subgroups.

## Proof

**Given**: , a left coset of in .

**To prove**: is contained in a left coset of in .

**Proof**: Since , any element in is also an element in . is a left coset of , so we see that is contained in a left coset of .

It's also clear that any left coset of that contains must be of the form , since it contains .