\(PSL(n,q)\) is obtained from determinant-one matrices over a finite field by factoring out scalar matrices. It is the most basic family of finite simple groups of Lie type. [@isaacs2008]
Definition
\(q\) is a prime power. The center has order \(gcd(n,q-1)\), so \(|PSL(n,q)|=|SL(n,q)|/gcd(n,q-1)\).
Core result
Elementary matrices generate \(SL(n,q)\) and establish perfectness in the stated range. Iwasawa's lemma combines the primitive projective action, perfectness, and a solvable normal subgroup of a point stabilizer to prove simplicity. See 8.29-8.33.
Structural properties
- \(PSL(2,2)\cong S_3\) and \(PSL(2,3)\cong A_4\), the nonsimple exceptions.
- \(PSL(2,4)\cong PSL(2,5)\cong A_5\).
- The projective action removes scalar matrices because they fix every one-dimensional subspace.
- Generation by elementary transvections underlies perfectness and simplicity.
- These groups form a classical family in the classification of finite simple groups.
Factoring the linear group by its center removes exactly the kernel of the projective action and joins matrix algebra to permutation primitivity.
Example and boundary
Knowledge network
The PSL-simplicity article concentrates the proof, permutation groups explain the projective action, and alternating groups supply low-order coincidences.
Proof and sources
Projective double transitivity, elementary generation, perfectness, and simplicity are Theorems 8.29-8.33. [@isaacs2008]
