# Subgroup containment implies coset containment

## Contents

DIRECT: The fact or result stated in this article has a trivial/direct/straightforward proof provided we use the correct definitions of the terms involved
View other results with direct proofs
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|

## 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 $K$.

## 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 $g$.