The simplicity theorem for \(PSL\) produces the first fundamental family of finite simple groups of Lie type. Its proof combines elementary-matrix generation, perfectness, and a primitive projective action. [@isaacs2008]
Definition
The low-order exceptions are \((n,q)=(2,2)\) and \((2,3)\).
Core result
Elementary transvections generate \(SL(n,q)\) and prove perfectness in the stated range. The central quotient acts doubly transitively on projective points. A point stabilizer has a solvable normal translation layer whose conjugates generate the group, so Iwasawa's lemma gives simplicity.
Structural properties
- \(PSL(2,2)\cong S_3\) and \(PSL(2,3)\cong A_4\) are nonsimple.
- Perfectness excludes nontrivial abelian quotients.
- Primitivity makes every nontrivial normal subgroup transitive.
- Conjugate generation by transvections excludes a merely local normal subgroup.
- Low-degree isomorphisms connect \(PSL(2,4)\) and \(PSL(2,5)\) with \(A_5\).
Iwasawa's lemma is a reusable template: a perfect primitive group generated by a solvable normal layer of a point stabilizer is simple.
Example and boundary
Knowledge network
The PSL article gives the construction, alternating groups supply low-order coincidences, and the simple-group article locates the family.
Proof and sources
Iwasawa's lemma, elementary generation, perfectness, and simplicity are Theorems 8.30-8.33; the proof is cited precisely here. [@isaacs2008]
