Projective special linear group

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Particular cases

Finite fields

Some facts:

Size of field Order of matrices Common name for the projective special linear group Order of group Comment
q 1 Trivial group 1 Trivial
2 2 Symmetric group:S3 6 = 2 \cdot 3 supersolvable but not nilpotent. Not simple.
3 2 Alternating group:A4 12 = 2^2 \cdot 3 solvable but not supersolvable group. Not simple.
4 2 Alternating group:A5 60 = 2^2 \cdot 3 \cdot 5 simple non-abelian group of smallest order.
5 2 Alternating group:A5 60 = 2^2 \cdot 3 \cdot 5 simple non-abelian group of smallest order.
7 2 Projective special linear group:PSL(3,2) 168 = 2^3 \cdot 3 \cdot 7 simple non-abelian group of second smallest order.
9 2 Alternating group:A6 360 = 2^3 \cdot 2^3 \cdot 5 simple non-abelian group.
2 3 Projective special linear group:PSL(3,2) 168 = 2^3 \cdot 3 \cdot 7 simple non-abelian group of second smallest order.
3 3 Projective special linear group:PSL(3,3) 5616 = 2^4 \cdot 3^3 \cdot 13 simple non-abelian group.

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