Finite Fields
The projective special linear groups PSL(n,Fq) for a finite field Fq are often written as PSL(n,q) or Ln(q). They are finite simple groups whenever n is at least 2, with two exceptions: L2(2), which is isomorphic to S3, the symmetric group on 3 letters, and is solvable; and L2(3), which is isomorphic to A4, the alternating group on 4 letters, and is also solvable.
The special linear groups SL(n,q) are thus quasisimple: perfect central extensions of a simple group (unless and or 3).
Read more about this topic: Projective Linear Group
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