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УДК 512.544.2
On Generation of the Group PSLn(Z + iZ) by Three Involutions, Two of Which Commute
Denis V.Levchuk*
Institute of Mathematics, Siberian Federal University, av. Svobodny 79, Krasnoyarsk, 660041,
Russia
Yakov N.Nuzhint
Institute of Fundamental Development, Siberian Federal University, st. Kirenskogo 26, Krasnoyarsk, 660074,
Russia
Received 20.01.2008, received in revised form 20.03.2008, accepted 05.04.2008 It is proved that the projective special linear group PSLn(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 PSLn(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 (2x2,2)-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 (2x2,2)-generated, then it will not be (2x2,2)-generated. Since there exist the homomorphism of PSLn(Z + iZ) onto PSLn(9) then the assertion of the Theorem 1 arises from the fact that the groups PSL2 (9) and PSL3(9) are not (2x2,2)-generated (see [1]). For n ^ 8 generating triples of involutions, two of which commute, of the group PSLn(Z + iZ) are indicated explicitly. It moreover n = 2(2k + 1) then we take generating triples of involutions from SLn(Z + iZ). Thus, for n ^ 8 and n = 2(2k + 1) we have a stronger statement: the group SLn(Z + iZ) is (2x2,2)-generated. Earlier, Ya.N.Nuzhin proved that PSLn(Z) is (2x2,2)-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 (2x2,2)-generation of the group SLn(Z) for n ^ 14.
*e-mail: dlevchuk82@mail.ru te-mail: nuzhin@fipu.krasnoyarsk.edu © Siberian Federal University. All rights reserved
1. Notations and Preliminary Results
Througout the paper Z are integers and Z + iZ are Gaussian integers, where i2 = -1. The rings Z and Z + iZ are Euclidean rings. Let R be an arbitrary Euclidean ring.
As usually, we will denote by tij(k), k G R, i = j, the transvections, that is, the matrices En + keij, where En is the identity (n x n) matrix, and ej denotes the (n x n) matrix with (i, j)-entry 1 and all other entries 0. The set tij(R) = {tij(k), k G R} is a subgroup. The following lemma is well-known (see, for example, ([4], p.107)).
Lemma 1. The group SLn(R) is generated by the subgroups tij(R), i,j = 1, 2,. .., n.
Let
/ 0 0 . .0 0 1 \ / 0 0 . .0 0 1 \
0 0.. .0 1 0 1 0 . .0 0 0
t = 0 0 . .1 0 0 , M = 0 1 . .0 0 0
\ 1 0.. .0 0 0 / \ 0 0 . .0 1 0 /
The matrix t is an involution, and the matrix m has the order n and acts regularly on the following set of subgroups:
M = {tin(R), ti+ii(R), i =1, 2,..., n - 1}.
Commuting among themselves subgroups of the set M, we can get all subgroups tij (R). Hence, by lemma 1 the group SLn(R) is generated by the set M. Moreover, the following lemma is true.
Lemma 2. The group SLn(R) is generated by one of the subgroups
tin(R), ti+ii(R), tn-in(R), tii+i(R), i = 1, 2, ...,n — 1,
and the monomial matrix for any diagonal matrix n with G SLn(R).
For elements of PSLn(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 SLn(R). In the next sections for elements of the groups SLn(R) and PSLn(R) we will also be using the teminology of Chevalley groups, consedering SLn(R) and PSLn(R) as universal and adjoint Chevalley group respectivelly.
Let $ be a root system of type Ai with the basis
n = {ri,r2,... ,ri},
where
l = n — 1.
The Chevalley group Ai(R) (universal and adjoint) of type Ai over ring R is generated of root subgroups
Xr = {xr(t), t G R}, r G $,
where xr (t) are root elements. For any r e $ and t = 0 we set
nr (t) = xr (t)x_r (—t-1)xr (t), nr = nr (1), hr ( —1) = n^
The map
tj+1j(t) ^ xri (t), i = 1, 2,...,/, t e R,
is extended up to isomorphism of the group SLn(R) onto universal Chevalley group Ai(R). The monomial matrices t and ^ indicated above are preimages of the elements wo and w respectively from the Weyl group W under natural homomorphism of the monomial subgroup N onto W, where wo(r) e $_ for any r e and
w = wri wr2 .. . wri.
Here $+ are the positive roots and $_ are negative roots. We can reformulate of Lemma 2 in terms of Chevalley groups.
Lemma 3. The Chevalley group Ai (R) is generated by any root subgroup
X±ri , ri e ^ X±(ri +---+r; )
and the monomial element nw if w = wri wr2 . .. wr;.
Througout the paper we use the notation ab = bab-1, [a, b] = aba-1b-1.
2. Generating Triples of Involutions
Let t and ^ be as in the first paragraph. The matrices t and
/ 0 0 . .0 1 0
0 0 . .1 0 0
1 0 . .0 0 0
V 0 0 . .0 0 1
are involutions, but they do not necessarity belong to SLn(Z) (this depends on their size). We select diagonal matrices ri1 and n2 with elements ±1 such that the matrices n1 t and t^ belong to SLn(Z) and their images in PSLn(Z) are involutions. We take ^1,V2 to be the following matrices:
for n = 4k + 1 (=5, 9,...)
m = V2 = En;
for n = 2(2k + 1) + 1 (=7,11,...)
m = -En, n2 = En;
for n = 4k (=8,12,...)
ni = E„, n2 = diag(E„_i,-1); for n = 2(2k + 1) (=6,10,...)
ni = diag(-#2fc+i, #2fc+i), n2 = E„.
Further for the group PSLn (Z + iZ) we explicitly write down triples of generating involutions a, P, y, two of which commute. For odd n ^ 7 by definition
a = i2i(«)in-in(i) diag(1, -1, -1, E„_e, -1, -1,1),
P = nir,
Y = n2T^.
For even n ^ 6 by definition
a = t2i(1)tn-in(-1) diag(1, -1, -1, En_6, -1, -1,1), P = diag(i, -i, 1,..., 1)niTdiag(-i, i, 1,..., 1),
Y = n2T«.
The next lemma is verified by direct calculation.
Lemma 4. Let a, P, y are search as above. Then:
1) aP = Pa;
2) a, y are involutions from SLn(Z + iZ) (and hence in PSLn(Z + iZ));
3) P is involution from SLn(Z + iZ) if n = 2(2k + 1);
4) if n = 2(2k + 1), then P2 = -En and hence image P is involution in PSLn(Z + iZ).
In Sections 3 and 4 we prove that involutions a, P, y generate the group PSLn(Z + iZ), n ^ 8, for odd and even n respectively. Further the next remark will be useful. By the construction
PY = n3M
for some diagonal element n3 € PSLn(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 a, P, y and contains one of the subgroups
tin(Z + iZ), tj+ij(Z + iZ), tn-in(Z + iZ), tjj+i(Z + iZ), i = 1, 2, ...,n - 1,
(in terminology of Chevalley groups one of root subgroups X±ri, r € n, X±(ri +_____+ri),) then
it coincides with the group PSLn(Z + iZ).
3. Proof of Theorem 1 for Odd n ^ 9
Let o, P, y, t, ni, n2, n3 be as in sections 1 and 2, n ^ 9 and l = n — 1. In terminology of Chevalley groups
a = xri (i)x_ri (i)hr2 ( —1)hr;_1 ( — 1),
P = ni t = nw0, Y = n2T^ = n„o nw,
n = Py = nw,
where w = wri wr2 ... wri.
Direct calculations give that
an = xr2 (±i)xri +-----(±i)hr3 ( —1)hri ( —1),
an2 = xr3 (±i)x_n (±i)hr4( —1)hri+...+r! ( —1),
[a, an] = xn+r2 (±1)xri+---+ri_i (±1), ([a, ° ]a^2 )2 = xri+r2+r3 (±i)xr2 (±i)xr2+r3 (11)xr2 + •••+ri _ i (Ii) e = (([a,an ]a^2 )2)n = xr2+r3 +r4 (±i)xr3 (±i)xr3+r4 ( ± 1)xr3 ++ •••■+ ri (Ii) [a,an]] = xn+r2+r3 (±i)xri+r2+r3+r4 (± 1)xri+•••+r; (Ii)
[a, [a, OLr>]]] = xri+-----+ ri_i (±1),
[a, [e, [a,an]]f = x_r2_____ri (±1),
[[e, [a, an]], [a, [e, [a, an]]]^] = xn (±i). Taking sequentially (l — 1)-commutator of the elements
^X r i ( I i), r i ( ± i) — ^Xy 2 ( I i), r 2 ( ± i) — r 3 ( I i), . . . , ^X r i i ( ± i) — ^Xy i ( I i ),
we get the element xri+_____+ri(±1). On the other hand, (xri(±i)n)^ = xri+_____+ri(±i).
All root subgroups Xr are commutative and, evidently, the elements 1 and i generate additively all ring Z + iZ. Therefore, if a subgroup is generated by involutions a, P, y, then it
contains the root subgroup Xri+_____+ri and hence by Lemma 5 it coincides with the subgroup
PSLn(Z + iZ).
Note, that for even n these generating involutions а, в, Y do not generate PSLn(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 PSLn(Z).
4. Proof of Theorem 1 for Even n ^ 8
Let а, в, Y,т, M, П1, П2, Пз be as in paragraphs 1 and 2, n ^ 8 and l = n — 1. In terminology of Chevalley groups
а = (1)x_n ( —1)hr2 ( — 1)hn-1 ( —1),
в = diag(i, —i, 1,..., 1)niT diag(—i, i, 1,..., 1) = hri (i)n„0(—i) = hri (i)hn (i)n„0, Y = П2ТМ = nw0 n„, П = eY = hri (i)hrj (i)nw, где w = wri wr2 ... wri.
Direct calculations give that
ап = жГ2 (±i)xn+...+n (±1)hr3 ( —1)hri ( —1),
2
ап = жГз (±i)x_n (±1)hr4 ( —1)hn+...+r! ( —1),
[а, ап] = Жп+Г2 (±i)xn+...+n_i (±1),
2 2
([а,аП]аП ) = Хп+Г2+Гз (±1)xr2 (±i)xr2+r3 (±1)xr2 + -+r;_i (±1),
в = (([а, ап]ап2 )2)n = жГ2+Гз+Г4 (±i)x^3 (±i)xr3+r4(±1)жГз+...+п (±i),
[в аП]] = Хп+Г2+Г3 (±1)xri+r2+r3+r4 (±i)xri + -+П (±^ [а, [в, [а, аП]]] = xri + -+r;_i (±1),
[а, [в, [а, ап]]]в = ж_Г2_____п (±1),
[[в, [а, ап]], [а, [в, [а, ап]]]в] = xri (±1). - 138 -
Taking sequentially (l — 1)-commutator of the elements xri (±1), xri (±1)n = xr2 (±i), xr2 (±i)n = xr3 (±i), ..., xri-3 (±i)n = xri_2 (±i),
Xn-2 (±i)n = xri-1 (±1), xri-1 (±1)n = xri (±1),
we get the element xri +_____+ri (±i). On the other hand, (xri (±1)n= xri+_____+ri (±1).
Therefore, if a subgroup is generated by the involutions a, 7, then it contains the root
subgroup Xri+_____+ri and hence by Lemma 5 it coincides with subgroup PSLn(Z + iZ).
This work has been supported by the RFFI Grant №07-01-00824.
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, 422-440 (Russian).
[2] Ya.N.Nuzhin, On generation of the group PSLn(Z) by three involutions, two of which commute, Vladikavkazskii math. journal, 10(2008), №1, 42-49 (Russian).
[3] M.C.Tamburini, Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute, J. of Algebra, 195(1997), №4, 650-661.
[4] R.Steinberg, Lectures on Chevalley groups, M.: Mir, 1975 (Russian).
