# Symmetric group S4

• Feb 19th 2013, 07:25 AM
awaisysf
Symmetric group S4
A=
 1 0 -1 -1
,B=
 0 √i i√i 0
,C=
 i 1 -i -1-i
,D=
 i√2 i(√i+√2) i√i -i√2

in above matrices "i" is iota ..

i want to prove that these four matrices generates S4 (symmetric group of order 24)
• Feb 19th 2013, 08:15 AM
HallsofIvy
Re: Symmetric group S4
Well, strictly speaking the don't. But they generate a group of matrices that is isomorphic to the set of permutations on four things (1234 if you wish). Can you find four such permutations that generate the group and then find a mapping from the set of matrices to those four permutations?
• Feb 19th 2013, 09:02 AM
awaisysf
Re: Symmetric group S4
when i take these matrices as mobius transformations and take action on projective line over the finite fields it gives S4, p = x^2+8y^2
, for example p=17
A=(0, 16, ∞) (1, 8, 15) (2, 11, 7) (3, 4, 10) (5, 14, 9) (6, 12, 13)
B=(0, ∞) (1, 13) (2, 15) (3, 10) (4, 16) (5, 6) (7, 14) (8) (9) (11, 12)
C=(0, 10, 4) (1, 7, 12) (2, 15, 9) (3, ∞, 16) (5, 8, 6) (11, 14, 13)
D=(0, 10) (1, 5) (2, 7) (3, ∞) (4, 16) (6, 13) (8, 9) (11) (12) (14, 15)
i want to give general prove..