Pairs of Generators for Matrix Groups. I

Transcription

1 Pairs of Generators for Matrix Groups I D E Taylor School of Mathematics Statistics The University of Sydney Australia 2006 It has been shown by Steinberg (962) that every finite simple group of Lie type can be generated by two elements These groups can be constructed from the simple Lie algebras over the complex numbers by the methods of Chevalley (955), Steinberg (959) Ree (96) The generators obtained by Steinberg (962) are given in terms of the root structure of the corresponding Lie algebras The identification of the groups of Lie type A n, B n, C n D n with classical matrix groups is due to Ree (957) an exposition of his results can be found in the book of Carter (972) The proofs ultimately rely on the work of Dickson (90) In this note we give tables of generators for the groups GL(n, q), SL(n, q), Sp(2n, q), U(n, q) SU(n, q) For the most part, the generators have been obtained by translating Steinberg s generators into matrix form via the methods of Ree (957) Notation Let E ij denote a square matrix with in the (i, j)th position 0 elsewhere For α GF (q) i j we set x ij (α) = I + αe ij we let h i (α) denote the diagonal matrix obtained by replacing the ith entry of the identity matrix by α The x ij (α) are the root elements of SL(n, q) Let w i denote the monomial matrix obtained from the permutation matrix corresponding to the transposition (i, i+) by replacing the (i+, i)-th entry by Then w = w w 2 w n represents the n-cycle (, 2,, n) Let be a generator of the multiplicative group of GF (q)

2 GL(n, q), q 2 Generators for GL(n, q) are Generators for Matrix Groups h () = x 2()w = When n = 2, the generators are ( ) 0 0 ( ) 0 Generators for GL(n, 2) = SL(n, 2) are given below 2 SL(n, q), q > 3 The generators are h ()h 2 ( ) = the matrix x 2 ()w given above 3 SL(n, 2) SL(n, 3) The generators are x 2 () = w =

3 4 Sp(2n, q), q odd, n > Generators for Matrix Groups The symplectic group Sp(2n, q) consists of the 2n 2n matrices X which satisfy the condition X t JX = J, where J = For i n, let i = 2n i + Define ĥ i (α) = h i (α)h i (α ) ˆx ij (α) = x ij (α)x j i ( α) Let ŵ be the monomial matrix obtained from the permutation matrix of the 2n-cycle (, 2,, n,, 2,, n ) by replacing the (2n, n)th entry by Generators for Sp(2n, q), q odd, are ĥ () = 3

4 ˆx 2 ()ŵ = When n = 2, the matrices are Note that Sp(2, q) SL(2, q) 5 Sp(2n, q), q even, q 2, n > The ˆx ij (α) are the short root elements of Sp(2n, q) The long root elements are transvections ẑ i (α) = x ii (α) Generators for Sp(2n, q) are ĥ ()ĥn() = 4

5 ˆx n ()ẑ ()ŵ = For Sp(4, q) the matrices become Sp(2n, 2), n > The generators are ˆx n ()ẑ () = ŵ =

6 7 Sp(4, 2) The matrices are U(2n, q), n > If x GF (q 2 ), we set x = x q If X is a matrix, X is obtained from X by replacing each entry x with x The unitary group consists of the matrices X such that X t JX = J, where J = Let be a primitive element of GF (q 2 ) let η be an element of trace 0, ie, η + η = 0 If q is odd, we may take η = (q+)/2 If q is even, we may take η = For this section define i = 2n + i set for i, j n h i (α) = h i (α)h i (α ), x ij (α) = x ij (α)x j i ( α), The matrix w is similar to ŵ of previous sections except that here it corresponds to the permutation (, 2,, n,, 2,, n ) it has η in the (, n + )th position η in the (2n, n)th position 6

7 Generators for U(2n, q) are h () = 0 η x 2 () w = η 0 η 0 When n = 2 the matrices are η η 0 0 η 0 0 7

8 9 SU(2n, q), n > Generators for Matrix Groups Generators are h () h 2 ( ) = x 2 () w 0 U(2n +, q) In this section let i = 2n + 2 i define h i (α) x ij (α) as before In the following matrices the boxed entry is in position (n +, n + ) For α, β GF (q 2 ) such that αα + β + β = 0, set I Q(α, β) = α β α Let w be the monomial matrix obtained from the permutation matrix of (n,, 2,, n,, 2, ) by replacing the (n +, n + )st entry by Let β be an element of GF (q 2 ) such that β + β = We may take β = ( + /) I 8

9 Generators are h n () = Q(, β)w = β SU(2n +, q), n or q 2 Generators are h n () h n+ ( ) = / Q(, β)w as above 9

10 2 SU(3, 2) Generators are References R W Carter, Simple Groups of Lie Type, (Wiley-Interscience: New-York 972) C Chevalley, Sur certaines groupes simples, Tôhoku Math J 7 (955), 4-66 L E Dickson, Linear groups, (Teubner: Leipzig 90) Rimhak Ree, On some simple groups defined by C Chevalley, Trans Amer Math Soc 84 (957), Rimhak Ree, A family of simple groups associated with the simple Lie algebras of type (F 4 ), Amer J Math 83 (96), Rimhak Ree, A family of simple groups associated with the simple Lie algebras of type (G 2 ), Amer J Math 83 (96), Robert Steinberg, Automorphisms of finite linear groups, Canad J Math 2 (960), Robert Steinberg, Generators for simple groups, Canad J Math 4 (962),