# Product of subsets whose total size exceeds size of group equals whole group

From Groupprops

## Statement

Suppose is a finite group and are (possibly equal, possibly distinct) subsets of such that . Then, every can be written as , where , . In other words, the product of subsets equals .

Note that this statement as given here applies *only* to the product of *two* subsets.

## Related facts

### Applications

- Every element of a finite field is expressible as a sum of two squares
- Subgroup of size more than half is whole group: This is usually proved in some other way, but it can be easily seen from this fact by taking the product of the subgroup with itself.

### Strictness

If , it is not necessary that . An example is when are both equal to each other and a subgroup of index two in .

### Similar facts

## Proof

**Given**: A finite group , (possibly equal, possibly distinct) subsets of such that . An element .

**To prove**: There exists such that .

**Proof**:

Step no. | Assertion/construction | Given data used | Previous steps used | Explanation |
---|---|---|---|---|

1 | The set is a subset of of the same size as . | inversion is one-one, and left multiplication by is one-one, so the composite is one-one. | ||

2 | is non-empty. Let be an element of the intersection. | Step (1) | By Step (1) and the given information, . By the principle of inclusion-exclusion, the subsets must intersect. | |

3 | for the obtained in Step (2). | Step (2) | Direct from Step (2), by multiplying both sides on the right by . |