# Subset containing identity whose left translates partition the group is a subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term subgroup

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## Statement

Suppose is a group and is a nonempty subset of containing the identity element of . Then, the following are equivalent:

- is a subgroup of
- For any , either or is empty
- For any , either or is empty

## Related facts

A reformulation, or immediate corollary, of this is the statement that:

Left congruence on a group equals left coset space relation from a subgroup

## Definitions used

### Subgroup

We shall use the definition of subgroup in terms of the subgroup criterion. A nonempty subset of a group is termed a subgroup if for any , the element is also in .

## Proof

### (1) implies (2) implies (3)

This is a direct consequence of the fact that for any subgroup, the left cosets partition the group.

### (3) implies (1)

**Given**: A group and nonempty subset such that for every either or is empty.

**To prove**: For any , the element is also in

**Proof**: Let . Consider the set . Since contains the identity element, , so is nonempty. Thus, by assumption, , so there exists such that , so . Thus, is in , completing the proof.