PSL(2,7) - Properties

Properties

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.

It is a general result that PSL(n, q) is simple for n ≥ 2, q ≥ 2 (q being some power of a prime number), unless (n, q) = (2,2) or (2,3). PSL(2, 2) is isomorphic to the symmetric group S₃, and PSL(2,3) is isomorphic to alternating group A₄. In fact, PSL(2,7) is the second smallest nonabelian simple group, after the alternating group A₅ = PSL(2,5) = PSL(2,4).

The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1,3,3,6,7,8.

Character table

$\begin{array}{r|cccccc} & 1A_{1} & 2A_{21} & 4A_{42} & 3A_{56} & 7A_{24} & 7B_{24} \\ \hline \chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \chi_2 & 3 & -1 & 1 & 0 & \sigma & \bar \sigma \\ \chi_3 & 3 & -1 & 1 & 0 & \bar \sigma & \sigma \\ \chi_4 & 6 & 2 & 0 & 0 & -1 & -1 \\ \chi_5 & 7 & -1 &-1 & 1 & 0 & 0 \\ \chi_6 & 8 & 0 & 0 & -1 & 1 & 1 \\ \end{array}, \sigma = \frac{-1+i\sqrt{7}}{2}.$

The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3,2), and the function notation for a representative in PSL(2,7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3,2) and PSL(2,7) can be switched arbitrarily.

Order	Size	Min Poly	Function
1	1	x+1	x
2	21	x2+1	−1/x
3	56	x3+1	2x
4	42	x3+x2+x+1	1/(3−x)
7	24	x3+x+1	x + 1
7	24	x3+x2+1	x + 3

The order of group is 168=3*7*8, this implies existence of Sylow's subgroups of orders 3, 7 and 8. It is easy to describe the first two, they are cyclic. Any element of conjugacy class 3A₅₆ generates Sylow 3-subgroup. Any element from the conjugacy classes 7A₂₄, 7B₂₄ generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral group of order 8. It can be described as centralizer of any element from the conjugacy class 2A₂₁. In the GL(3,2) representation, a Sylow 2-subgroup consists of the upper triangular matrices.

This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for p=2.

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