Journal of Siberian Federal University. Mathematics & Physics 2 (2008)

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1 Journal of Siberian Federal University. Mathematics & Physics 2 (2008) УДК On Generation of the Group PSL n (Z + iz) by Three Involutions, Two of Which Commute Denis V.Levchuk Institute of Mathematics, Siberian Federal University, av. Svobodny 79, Krasnoyarsk, , Russia Yakov N.Nuzhin Institute of Fundamental Development, Siberian Federal University, st. Kirenskogo 26, Krasnoyarsk, , Received , received in revised form , accepted Russia It is proved that the projective special linear group P SL n(z + iz), n 8, over Gaussian integers Z + iz is generated by three involutions, two of which commute. Keywords: gaussian intergers, special linear group, generating elements. Introduction The main result of the paper is the following theorem. Theorem 1. For n 8 the projective special linear group P SL n (Z + iz) over Gaussian integers Z + iz is generated by three involutions, two of which commute, but for n = 2, 3 it is not generated by such three involutions. The groups generated by three involutions, two of which commute, will be called (2 2,2)- generated. Here we do not exclude the cases when two or even three involutions are the same. Clearly, if a group has a homomorphic image, which is not (2 2,2)-generated, then it will not be (2 2,2)-generated. Since there exist the homomorphism of P SL n (Z + iz) onto P SL n (9) then the assertion of the Theorem 1 arises from the fact that the groups P SL 2 (9) and P SL 3 (9) are not (2 2,2)-generated (see [1]). For n 8 generating triples of involutions, two of which commute, of the group P SL n (Z + iz) are indicated explicitly. It moreover n 2(2k + 1) then we take generating triples of involutions from SL n (Z + iz). Thus, for n 8 and n 2(2k + 1) we have a stronger statement: the group SL n (Z + iz) is (2 2,2)- generated. Earlier, Ya.N.Nuzhin proved that P SL n (Z) is (2 2,2)-generated if and only if n 5. In the proof of Theorem 1 the methods of choosing generating triples of involutions developed in [2] are essentially used. Note also that M.C.Tamburini and P.Zucca [3] proved (2 2,2)-generation of the group SL n (Z) for n dlevchuk82@mail.ru nuzhin@fipu.krasnoyarsk.edu c Siberian Federal University. All rights reserved 133

2 1. Notations and Preliminary Results Througout the paper Z are integers and Z + iz are Gaussian integers, where i 2 = 1. The rings Z and Z + iz are Euclidean rings. Let R be an arbitrary Euclidean ring. As usually, we will denote by t ij (k), k R, i j, the transvections, that is, the matrices E n + ke ij, where E n is the identity (n n) matrix, and e ij denotes the (n n) matrix with (i, j)-entry 1 and all other entries 0. The set t ij (R) = {t ij (k), k R} is a subgroup. The following lemma is well-known (see, for example, ([4], p.107)). Lemma 1. The group SL n (R) is generated by the subgroups t ij (R), i, j = 1, 2,..., n. Let τ = , µ = The matrix τ is an involution, and the matrix µ has the order n and acts regularly on the following set of subgroups: M = {t 1n (R), t i+1i (R), i = 1, 2,..., n 1}. Commuting among themselves subgroups of the set M, we can get all subgroups t ij (R). Hence, by lemma 1 the group SL n (R) is generated by the set M. Moreover, the following lemma is true. Lemma 2. The group SL n (R) is generated by one of the subgroups t 1n (R), t i+1i (R), t n 1n (R), t ii+1 (R), i = 1, 2,..., n 1, and the monomial matrix ηµ for any diagonal matrix η with ηµ SL n (R). For elements of P SL n (R) we will be also using matrix representation, assuming that two element are equal if they only differ by multiplication with a scalar matrix of SL n (R). In the next sections for elements of the groups SL n (R) and P SL n (R) we will also be using the teminology of Chevalley groups, consedering SL n (R) and P SL n (R) as universal and adjoint Chevalley group respectivelly. Let Φ be a root system of type A l with the basis Π = {r 1, r 2,..., r l },. where l = n 1. The Chevalley group A l (R) (universal and adjoint) of type A l over ring R is generated of root subgroups X r = {x r (t), t R}, r Φ, 134

3 where x r (t) are root elements. For any r Φ and t 0 we set n r (t) = x r (t)x r ( t 1 )x r (t), n r = n r (1), h r ( 1) = n 2 r. The map t i+1i (t) x ri (t), i = 1, 2,..., l, t R, is extended up to isomorphism of the group SL n (R) onto universal Chevalley group A l (R). The monomial matrices τ and µ indicated above are preimages of the elements w 0 and w respectively from the Weyl group W under natural homomorphism of the monomial subgroup N onto W, where w 0 (r) Φ for any r Φ + and w = w r1 w r2... w rl. Here Φ + are the positive roots and Φ are negative roots. We can reformulate of Lemma 2 in terms of Chevalley groups. Lemma 3. The Chevalley group A l (R) is generated by any root subgroup X ±ri, r i Π, X ±(r1+ +r l ) and the monomial element n w if w = w r1 w r2... w rl. Througout the paper we use the notation a b = bab 1, [a, b] = aba 1 b Generating Triples of Involutions Let τ and µ be as in the first paragraph. The matrices τ and τµ = are involutions, but they do not necessarity belong to SL n (Z) (this depends on their size). We select diagonal matrices η 1 and η 2 with elements ±1 such that the matrices η 1 τ and η 2 τµ belong to SL n (Z) and their images in P SL n (Z) are involutions. We take η 1, η 2 to be the following matrices: for n = 4k + 1 (= 5, 9,... ) η 1 = η 2 = E n ; for n = 2(2k + 1) + 1 (= 7, 11,... ) η 1 = E n, η 2 = E n ; 135

4 for n = 4k (= 8, 12,... ) η 1 = E n, η 2 = diag(e n 1, 1); for n = 2(2k + 1) (= 6, 10,... ) η 1 = diag( E 2k+1, E 2k+1 ), η 2 = E n. Further for the group P SL n (Z + iz) we explicitly write down triples of generating involutions α, β, γ, two of which commute. For odd n 7 by definition α = t 21 (i)t n 1n (i) diag(1, 1, 1, E n 6, 1, 1, 1), β = η 1 τ, γ = η 2 τµ. For even n 6 by definition α = t 21 (1)t n 1n ( 1) diag(1, 1, 1, E n 6, 1, 1, 1), β = diag(i, i, 1,..., 1)η 1 τ diag( i, i, 1,..., 1), γ = η 2 τµ. The next lemma is verified by direct calculation. Lemma 4. Let α, β, γ are search as above. Then: 1) αβ = βα; 2) α, γ are involutions from SL n (Z + iz) (and hence in P SL n (Z + iz)); 3) β is involution from SL n (Z + iz) if n 2(2k + 1); 4) if n = 2(2k + 1), then β 2 = E n and hence image β is involution in P SL n (Z + iz). In Sections 3 and 4 we prove that involutions α, β, γ generate the group P SL n (Z + iz), n 8, for odd and even n respectively. Further the next remark will be useful. By the construction βγ = η 3 µ for some diagonal element η 3 P SL n (Z + iz). Therefore by Lemma 2 the proof of Theorem 1 can be reduced to verification of the hypothesis of the next lemma. Lemma 5. If a group is generated by involutions α, β, γ and contains one of the subgroups t 1n (Z + iz), t i+1i (Z + iz), t n 1n (Z + iz), t ii+1 (Z + iz), i = 1, 2,..., n 1, (in terminology of Chevalley groups one of root subgroups X ±ri, r i Π, X ±(r1+ +r l ),) then it coincides with the group P SL n (Z + iz). 136

5 3. Proof of Theorem 1 for Odd n 9 Let α, β, γ, τ, µ, η 1, η 2, η 3 be as in sections 1 and 2, n 9 and l = n 1. In terminology of Chevalley groups α = x r1 (i)x rl (i)h r2 ( 1)h rl 1 ( 1), β = η 1 τ = n w0, γ = η 2 τµ = n w0 n w, η βγ = n w, where w = w r1 w r2... w rl. Direct calculations give that α η = x r2 (±i)x r1+ +r l (±i)h r3 ( 1)h rl ( 1), α η2 = x r3 (±i)x r1 (±i)h r4 ( 1)h r1+ +r l ( 1), [α, α η ] = x r1+r 2 (±1)x r1+ +r l 1 ([α, α η ]α η2 ) 2 = x r1+r 2+r 3 (±i)x r2 (±i)x r2+r 3 (±1)x r2+ +r l 1 θ (([α, α η ]α η2 ) 2 ) η = x r2+r 3+r 4 (±i)x r3 (±i)x r3+r 4 (±1)x r3+ +r l [θ, [α, α η ]] = x r1+r 2+r 3 (±i)x r1+r 2+r 3+r 4 (±1)x r1+ +r l [α, [θ, [α, α η ]]] = x r1+ +r l 1 [α, [θ, [α, α η ]]] β = x r2 r l [[θ, [α, α η ]], [α, [θ, [α, α η ]]] β ] = x r1 (±i). Taking sequentially (l 1)-commutator of the elements x r1 x r1 (±i) η = x r2 x r2 (±i) η = x r3..., x rl 1 (±i) η = x rl we get the element x r1+ +r l (±1). On the other hand, (x rl (±i) η ) β = x r1+ +r l (±i). All root subgroups X r are commutative and, evidently, the elements 1 and i generate additively all ring Z+iZ. Therefore, if a subgroup is generated by involutions α, β, γ, then it contains the root subgroup X r1+ +r l and hence by Lemma 5 it coincides with the subgroup P SL n (Z + iz). 137

6 Note, that for even n these generating involutions α, β, γ do not generate P SL n (Z + iz), since by conjugating by diagonal element diag(1, i, 1, 1, i, 1, 1,..., i, 1, 1, i) the subgroup generated by these involutions, we only get the group P SL n (Z). 4. Proof of Theorem 1 for Even n 8 Let α, β, γ, τ, µ, η 1, η 2, η 3 be as in paragraphs 1 and 2, n 8 and l = n 1. In terminology of Chevalley groups α = x r1 (1)x rl ( 1)h r2 ( 1)h rl 1 ( 1), β = diag(i, i, 1,..., 1)η 1 τ diag( i, i, 1,..., 1) = h r1 (i)n w0 h r1 ( i) = h r1 (i)h rl (i)n w0, γ = η 2 τµ = n w0 n w, η βγ = h r1 (i)h rl (i)n w, где w = w r1 w r2... w rl. Direct calculations give that α η = x r2 (±i)x r1+ +r l (±1)h r3 ( 1)h rl ( 1), α η2 = x r3 (±i)x r1 (±1)h r4 ( 1)h r1+ +r l ( 1), [α, α η ] = x r1+r 2 (±i)x r1+ +r l 1 ([α, α η ]α η2 ) 2 = x r1+r 2+r 3 (±1)x r2 (±i)x r2+r 3 (±1)x r2+ +r l 1 θ (([α, α η ]α η2 ) 2 ) η = x r2+r 3+r 4 (±i)x r3 (±i)x r3+r 4 (±1)x r3+ +r l [θ, [α, α η ]] = x r1+r 2+r 3 (±1)x r1+r 2+r 3+r 4 (±i)x r1+ +r l [α, [θ, [α, α η ]]] = x r1+ +r l 1 [α, [θ, [α, α η ]]] β = x r2 r l [[θ, [α, α η ]], [α, [θ, [α, α η ]]] β ] = x r1 (±1). 138

7 Taking sequentially (l 1)-commutator of the elements x r1 x r1 (±1) η = x r2 x r2 (±i) η = x r3..., x rl 3 (±i) η = x rl 2 x rl 2 (±i) η = x rl 1 x rl 1 (±1) η = x rl we get the element x r1+ +r l (±i). On the other hand, (x rl (±1) η ) β = x r1+ +r l (±1). Therefore, if a subgroup is generated by the involutions α, β, γ, then it contains the root subgroup X r1+ +r l and hence by Lemma 5 it coincides with subgroup P SL n (Z + iz). This work has been supported by the RFFI Grant References [1] Ya.N.Nuzhin, Generating triples of involutions of the groups of Lie type over finite field of odd characteristic. II, Algebra i Logika, 36(1997), 4, (Russian). [2] Ya.N.Nuzhin, On generation of the group P SL n (Z) by three involutions, two of which commute, Vladikavkazskii math. journal, 10(2008), 1, (Russian). [3] M.C.Tamburini, Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute, J. of Algebra, 195(1997), 4, [4] R.Steinberg, Lectures on Chevalley groups, M.: Mir, 1975 (Russian). 139