# Simple and non-abelian implies perfect

From Groupprops

## Contents

## Statement

Suppose is a simple non-abelian group. Then, is a perfect group, i.e., equals its own derived subgroup.

## Related facts

## Facts used

## Proof

**Given**: A simple non-abelian group .

**To prove**: The derived subgroup equals .

**Proof**:

No. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|

1 | The derived subgroup of is normal in . | Fact (1) | Fact-direct. | ||

2 | The derived subgroup of is either the trivial subgroup or the whole group . | is simple. | Step (1) | Step-given direct, combined with the definition of simple. | |

3 | The derived subgroup of cannot be the trivial subgroup. | is non-abelian. | By definition, the derived subgroup is trivial if and only if the group is abelian. | ||

4 | The derived subgroup of is the whole group . | Steps (2), (3) | Step-combination direct. |