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 $H\leq K\leq G$ are subgroups. Then, every Left coset (?) of $H$ is contained in exactly one left coset of $K$.

For right cosets: Suppose $H\leq K\leq G$ are subgroups. Then, every Right coset (?) of $H$ is contained in exactly one right coset of $K$.

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: $H\leq K\leq G$, a left coset $gH$ of $H$ in $G$.

To prove: $gH$ is contained in a left coset of $K$ in $G$.

Proof: Since $H\leq K$, any element in $gH$ is also an element in $gK$. $gK$ is a left coset of $K$, so we see that $gH$ is contained in a left coset of $K$.

It's also clear that any left coset of $K$ that contains $gH$ must be of the form $gK$, since it contains $g$.