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Generation of the Chevalley Group of Type G2 over the Ring of Integers by Three Involutions Two of which Commute
It is proved that G2 (Z) is generated by three involutions. Two of these involutions commute. Keywords: ring of integers, generating involutions, Chevalley group
Introduction
The main result of this article is
Theorem 1. The Chevalley group G2(Z) over the ring of integers Z is generated by three involutions and two of these involutions commute.
Theorem 1 answers the question formulated by Ya. N.Nuzhin [1, question 15.67] for the group G2(Z) : What adjoint Chevalley groups over the ring of integers are generated by three involutions, two of which commute?
This problem has not been solved. We just know that groups SLn(Z), n > 14 are generated by three involutions, two of which commute [2]. Groups PSLn(Z) are generated by three involutions, two of which commute when n > 5 [3]. Note also that adjoint Chevalley group B2(Z) is not generated by three involutions, two of which commute. It follows from the fact that group PSp4(3) is not generated by three involutions, two of which commute [4].
1. Notation and preliminary results
Let $ be a reduced indecomposable root system. Let us denote adjoint Chevalley group over a field K by $(K). This group is generated by root subgroups Xr = {xr(t) | t e K}, r e Let us denote special linear group by SL2(K) and subgroup generated by the set M by (M}.
Lemma 1 ( [5, Theorem 6.3.1., p.88]). There is a homomorphism from SL2(K) onto subgroup (Xr ,X-r} of $(K) such that
Ivan A. Timofeenko*
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
*
Received 10.12.2014, received in revised form 21.12.2014, accepted 24.01.2015
* ivan.timofeenko@gmail.com © Siberian Federal University. All rights reserved
Then K* is the multiplicative group of the field K. Let us assume that
nr (t) = xr (t)x_r (—t-1)xr (t), hr (t) = nr (t)nr ( —1),
nr = nr(1), r e t e K*.
With conjugations the diagonal elements act on root elements as follows:
hr (t)xs(u)hr (t)-1 = xs(tArs u), (1)
where Ars = 2(r, s)/(r, r) and (x, y) is the scalar product of vectors x, y.
Let H be a diagonal subgroup of a group $(K) generated by elements hr(t), r e $,t e K*. Let N be a monomial subgroup of the group $(K) generated by H and elements nr, r e $ and let W be a Weyl group of type
Lemma 2. The Chevalley group $(R) over an euclidian ring R is generated by the root elements xr(1), r e ±n, where n is a fundamental root subsystem of the root system
Proof. Let us assume that G = (xr (1) | r e ±n) then nr e G for r e ±n. Fundamental reflections wr, r e n are images of elements nr, r e n under homomorphism from N into W. Elements wr, r e n generate the group W [5, Proposition 2.1.8, p. 17]. Hence ns e G for all s e Since
nsxr (t)n-1 = x№s(r)(±t)
and group W acts transitivly on the roots with the same length then xr (1) e G for all r e By consequence 3 from [6, c. 107] group $(K) is generated by root elements xr (1), r e hence G = $(K). □
Next we need 7-dimensional matrix representation of the Chevalley group G2(K) [7]. Let us fix a fundamental system of roots of (a, b} of type G2. Then the root elements have the following representation
xa(t) = e + t(e67 + 2e45 - e34 - ei2) - t2e35, x_a(t) = e + t(e76 + e54 - 2e43 - e2i) - t2e53, xa+b(t) = e + t(ei3 - e24 + 2e46 - e5r) - t2e26, x_a_b(t) = e + t(e3i - 2e42 + e64 - e75) - t2e62, x2a+b(t) = e + t(e47 + e36 - e25 - 2ew) - t2ei7, x_2a_b(t) = e + t(e74 + e63 - e52 - 2e4i) - t2e7i,
x6(t) = e + t(e56 - e23), x_b(t) = e + t(e65 - e32),
x3a+b(t) = e + t(ei5 - e37),
x_3a_b(t) = e + t(e5i - e73),
x3a+2b(t) = e + t(e27 - ei6),
x_3a_2b(t) = e + t(e72 - e6i),
where e is the indentity matrix, matrices ej have entries equal to 1 at (i,j) and other entries equal to 0.
2. Proof of Theorem 1
As in previous section (a, b} is a fundamental root system of type G2, where a is the short root.
Let us denote
a = x0(1)h6(-1), 3 = x_6(1)ha(-1),
Y = n0n30+26h6(-1).
Our goal is to show that (a, 3, y} are three involutions that generate group G2(Z) with
a3 = ¡a.
Let us show that a, 3 and y are involutions. Applying equality (1), we obtain a2 = xa(1)h6(-1)xa(1)h6(-1) = xa(1)xa((-1)Aba) = 1,
because A = 2M = -2^bIH = 1
because Aba = (b,b) = 2|b||b| =1.
Similarly we have
32 = X_6(1)ha(-1)x_6(1)ha(-1) = X-6(1)X-6((-1)Aa--b ) = 1,
because Aa_6 = 2(a b) = 2ffiallb| = 3.
(a, a) 2|a||a|
Elements na and n3a+2b are commute because a ± (3a + 2b) is not a root. Hence we have
2
Y = na n3a+26«a«3a+26 =
= nanan3a+2b «3a+26 = = ha(-1)h3a+2b(-1) = 1.
Now we show that the equality above is true. Diagonal elements ha(-1) and h3a+2b(-1) act equally by conjugations (1) on generating elements xa(1) and xb(1) of group G2(Z). Note also that elements ha(-1) and h3a+2b(-1) from matrix representation of group G2(K) over field K [7] are represented by matrix diag(-1, -1,1,1,1, -1, -1).
Then we show that a3 = 3a. For this we just need to show that (a3)2 = 1. Simple manipulations give us the following result
(a3)2 = Xa(1)h6(-1)x_6 (1)ha(-1)Xa(1)M-1)X-b(1)ha(-1) =
= h6(1)ha(1)xa(-1)x_6(-1)xa (1)x_6(1)h6(-1)ha(-1) =
= h6(1)ha(1)xa(-1)xa(1)x_6(-1)x_6(1)h6 (-1)ha(-1) =
= h6(1)ha(1)h6( 1)ha( 1) = = h6(1)h6(-1)ha(1)ha(-1) = 1.
Let us denote M = (a, 3, y). We show that M = G2(Z). We have the following relation
a7 = nan3a+26h6(-1)Xa(1)h6(-1)nan3a+26h6 (-1) =
2(b,a)
= na«3a+26Xa((-1) (b,b) )nan3a+2bh6(-1) =
2^3|q||b| (2)
= «a«3a+26Xa((-1) 2|a||a| )nan3a+2bh6(-1) = (2)
= «a«3a+26Xa((-1)1)nan3a+26h6(-1) =
= n3a+26naXa(-1)nan3a+26h6(-1).
In matrix representation of the group G2(Z) we have
n3a+2b
0 -1 0 0 0 0 0
1 0 0 0 0 0 0
0 0 0 0 -1 0 0
0 0 0 -1 0 0 0
0 0 -1 0 0 0 0
0 0 0 0 0 0 1
0 0 0 0 0 -1 0
0 0 00 0 -1 0 \
0 0 00 0 0 1
0 0 10 0 0 0
2b = 0 0 01 0 0 0
0 0 00 1 0 0
1 0 00 0 0 0
0 -1 00 0 0 0
( 1 1 00 0 0 0 \
0 1 00 0 0 0
0 0 11 -1 0 0
1) = 0 0 01 -2 0 0
0 0 00 1 0 0
0 0 00 0 1 -1
0 0 00 0 0 1
Following some manipulations we obtain
n3a+2b na xa(-1)na n3a+2b
/ 1 0 0 0 0 0 0
-1 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 -2 1 0 0 0
0 0 -1 1 1 0 0
0 0 0 0 0 1 0
V 0 0 0 0 0 1 1
x_a(1).
Thus, in view of (2) we have
Let us introduce
= x_a(1)h6(-1).
9 = aaY = xa(1)x_a( — 1).
We show that 93 = ha(-1). Since mapping — from Lemma 1 is isomorphism for group G2(Z) we can use matrix representation. Then manipulations with matrices of the second order give the following equalities
93 = (xa(1)x_a (-1))3
0 1 11
1 1 1 0 0 1 1 1
-1 0 01
= ha(-1).
n
a
x
a
Y
a
3
Therefore
ha(-1) G M. (3)
Then
в7 = xb(1)ha(-1). (4)
It follows from (4) that xb(±1) = в7ha(-1). After applying (3), we get inclusion
x6(±1) G M.
By definition nb = (1)x_b(- 1)xb(1). Then x_b(±1) = (1))Y and nb G M. Therefore
n2 = h6(-1) G M. (5)
From relation (5) and equality xa(1) = ahb(-1) we get inclusion
xa(1) G M.
The ring of integers Z is euclidean ring then by Lemma 2 and inclusions
x±a(1), x±6(1) G M,
we obtain M = G2(Z).
Therefore, group G2(Z) is generated by three involutions а, в and 7. First two involutions commute. □
The author thanks Ya. N. Nuzhin for problem formulation and attention to the work.
References
[1] The Kourovka Notebook. Unsolved problems in group theory, 17th edition, 2010 (in Russian).
[2] M.C.Tamburini, P.Zucca, Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute, J. of Algebra, 195(1997), 650-661.
[3] Ya.N.Nuzhin, On Generation of Groups PSLn(Z) by Three Involutions, Two of Which Commute, Vladikavkaz. Мatemat. Zh., 10(2008), no. 1, 68-74 (in Russian).
[4] 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), no. 4, 422-440 (in Russian).
[5] R.W.Carter, Simple Groups of Lie Type, John Wiley and Sons, 1972.
[6] R.Steinberg, Lectures on Chevalley groups, Yale Univ., Math. Dept. 1967.
[7] V.M.Levchuk, Ya.N.Nuzhin, Structure of Ree groups, Algebra i Logika, 24(1985), no. 1, 26-41 (in Russian).
Порождающие тройки инволюций группы Шевалле типа G2 над кольцом целых чисел
Иван А. Тимофеенко
В 'работе доказано, что группа G2(Z) порождается тремя инволюциями, две из которых перестановочны.
Ключевые слова: кольцо целых чисел, порождающие тройки инволюций, группа Шевалле
