Simple and non-abelian implies perfect

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Statement

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

Related facts

Facts used

  1. Derived subgroup is normal

Proof

Given: A simple non-abelian group G.

To prove: The derived subgroup [G,G] equals G.

Proof:

No. Assertion/construction Facts used Given data used Previous steps used Explanation
1 The derived subgroup of G is normal in G. Fact (1) Fact-direct.
2 The derived subgroup of G is either the trivial subgroup or the whole group G. G is simple. Step (1) Step-given direct, combined with the definition of simple.
3 The derived subgroup of G cannot be the trivial subgroup. G is non-abelian. By definition, the derived subgroup is trivial if and only if the group is abelian.
4 The derived subgroup of G is the whole group G. Steps (2), (3) Step-combination direct.