Simple and non-abelian implies perfect
Given: A simple non-abelian group .
To prove: The derived subgroup equals .
|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.|