Nonempty intersection of cosets is coset of intersection
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If the intersection of a collection of left cosets of subgroups is nonempty, then it is a coset of the intersection of the corresponding subgroups.
Statement with symbols
Suppose is a family of subgroups of a group indexed by , and are elements of . Then , if non-empty, is a left coset of the subgroup .
- Subgroup containment implies coset containment: 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.
- 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.
Given: , , non-empty.
To prove: there exists such that
Proof: Observe that for any in , we have , viz: . So, .
For any , in each we can find such that . Therefore is in . Hence is in , viz: .
Now, for all in , we can find such that and . Then . So it follows that the cosets of the intersection subgroup with respect to are the same. Therefore, . Hence .