Coset containment implies subgroup containment
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If a Left coset (?) of one subgroup is contained in a left coset of another subgroup, then the subgroup is also contained in the other.
Statement with symbols
Suppose are cosets of subgroups . Then .
- 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.
- 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.
Given: for some .
Proof: Since is in , which is contained in , there exists such that . Thus , which implies that there exist and such that . Hence , which is in . So, .