# Equivalence of Definitions of Complement of Subgroup

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## Contents

## Theorem

Let $G$ be a group with identity $e$.

Let $H$ and $K$ be subgroups.

The following definitions of the concept of **Complement of Subgroup** are equivalent:

### Definition $1$

$K$ is a **complement** of $H$ if and only if:

- $G = H K$ and $H \cap K = \set e$

### Definition $2$

$K$ is a **complement** of $H$ if and only if:

- $G = K H$ and $H \cap K = \set e$

## Proof

### Definition $1$ implies Definition $2$

Let $G = H K$.

Then $H K$ is a group.

By Subset Product of Subgroups:

- $H K = K H$

Thus $K H = G$.

$\Box$

### Definition $2$ implies Definition $1$

Let $G = K H$.

Then $K H$ is a group.

By Subset Product of Subgroups:

- $H K = K H$

Thus $H K = G$.

$\blacksquare$