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

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

Related facts

Proof

Given: H \le K \le 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 \le 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.