PSL(2,7)
In mathematics, the projective special linear group PSL(2, 7) (isomorphic to GL(3, 2)) is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements PSL(2, 7) is the second-smallest nonabelian simple group after the alternating group A5 on five letters with 60 elements (the rotational icosahedral symmetry group), or the isomorphic PSL(2, 5).
The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have nonzero determinant. The subgroup SL(2, 7) consists of all such matrices with unit determinant. Then PSL(2, 7) is defined to be the quotient group
obtained by identifying I and −I, where I is the identity matrix. In this article, we let G denote any group isomorphic to PSL(2, 7).
G = PSL(2, 7) has 168 elements. This can be seen by counting the possible columns; there are 72−1 = 48 possibilities for the first column, then 72−7 = 42 possibilities for the second column. We must divide by 7−1 = 6 to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is (48×42)/(6×2) = 168.
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