On intersection of primary subgroups in the group Aut(F4(2)) Текст научной статьи по специальности «Математика»

УДК 512.542

On Intersection of Primary Subgroups in the Group Aut(F4(2))

Viktor I. Zenkov*

Institute of Mathematics and Mechanics UB RAS Kovalevskoi, 16, Ekaterinburg, 620990 Ural Federal University Mira, 19, Ekaterinburg, 620990 Russia

Yakov N. Nuzhint

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 20.05.2017, received in revised form 29.12.2017, accepted 20.01.2018 It is proved that, in a finite group G which is isomorphic to the group of automorphisms of the Chevalley group F4 (2), there are only three possibilities for ordered pairs of primary subgroups A and B with condition: A n Bg = 1 for any g £ G. We describe all ordered pairs (A, B) of such subgroups up to conjugacy in the group G and in particular, we prove that A and B are 2-groups.

Keywords: finite group, almost simple group, primary subgroup. DOI: 10.17516/1997-1397-2018-11-2-171-177.

1. Introduction and preliminaries

Let G be a finite group and A and B be its subgroups. By definition, M is the set of subgroups that are minimal by inclusion among all subgroups of type A n Bg, g e G, and m consists of those elements of the set M whose order is minimal. Denote by MinG(A, B) (resp. minG(A, B)) the subgroup, generated by the set M (resp. m). First this kind of groups was introduced in [1]. Evidently, minG(A, B) ^ MinG(A, B) and the following three conditions are equivalent: a) A n Bg = 1 for any g e G; b) MinG(A, B) = 1; c) minG(A, B) = 1.

If S e Sylp(G) then subgroups minG(S, S) = 1 can be described in many interesting cases. It give us a description of pairs of subgroups (A, B) with the condition minG(A, B) = 1 for primary subgroups and sometimes for nilpotent subgroups A and B. For example, in [2, Theorem 1] it is proved that MinG(A, B) < F(G) for any pair of abelian subgroups A and B of G, where F(G) is the Fitting subgroup of G (the greatest normal nilpotent subgroup of G).

It was proved in [3] that if G is an almost simple group with socle L2 (q), q > 3, and S e Sylp(G), then minG(S, S) = MinG(S, S) = S for the Mersenne prime q = 2n - 1, and the equalities minG(S, S) = MinG(S, S) = 1 hold for all others q, exception q = 9. For q = 9

* [email protected] 1 [email protected] © Siberian Federal University. All rights reserved

the subgroup minG(S, S) is isomorphic to the dihedral group Di6 and it has index 2 in the group S. The exceptional case is important for our paper, therefore we mention corresponding result of [4].

Let socle of G be isomorphic to L2(9) ~ A6 and S G Sylp(G). Then minG(S,S) = 1 for p > 2, but for p = 2 the equality minG(S, S) = 1 holds for all G, exception G = Aut(A6). In exception case minG(S, S) = (i,j) ~ Di6, i2 = j2 = 1, and (i)| = (j)| = 8, where i,j belongs to S \ S n Soc(G) and j (resp. i) induces field (resp. diagonal) automorphism of the group Soc(G).

We need some information about subgroups of the Chevalley group F4(2). Let r2 and r3 be fundamental roots of the root system of type F4 which generate subsystem of type B2. Denote by P{2<3} the parabolic subgroup which is generated by monomial elements nr2, nr3 and unipotent subgroup U corresponding positive roots. The subgroup P{2<3} is invariant under graph automorphism t of order 2, its Levy subgroups L is isomorphic to the Chevalley group B2(2). The product L(t) is isomorphic to the group Aut(A6). We prove the following theorems.

Theorem 1. Let G be a finite group with socle F4(2), S be a Sylow 2-subgroup of G and MinG(S,S) = 1. Then G ~ Aut(F4(2)) and

minG(S, S) = O2(P{2,3}) • minL{T)(S1,S1),

where Si is a Sylow 2-subgroup of the group L(t) and minL^T)(Si, Si) ~ Di6.

Theorem 2. Let A, B be p-subgroups of a finite group G with socle F4(2) and S be a Sylow 2-subgroup of G. Then the following are equivalent:

1) MinG(A,B) = 1;

2) p = 2, G ~ Aut(F4(2)) and up to conjugacy in the group G the ordered pair (A,B) lies in the set {(S, S), (minG(S, S),S), (S, minG(S, S))}.

2. Notations and preliminary results

Further, G be a finite group, A and B be its subgroups. The sets M, m and the subgroups MinG(A, B), minG(A, B) as in the introduction. Others notations are standard for group theory. For example, Sylp(G) is the set of all Sylow p-subgroups of the group G, and Soc(G) is the socle of G (the minimal normal subgroup of the group G).

Lemma 2.1 ( [4]). Let Soc(G) = A6, S G Sylp(G) and minG(S,S) = 1. Then G = Aut(A6), p = 2 and minG(S, S) = (i,j) ~ Dl6, where i2 = j2 = 1, |Cs(i)| = |Cs(j)| = 8, the order of each elements of m is equal to 2 and the subgroup (i,j) cowers quotient groups G/G'.

Lemma 2.2 ([4]). Let A, B be p-subgroups of G, Soc(G) = A6, S G Sylp(G) and minG(A, B) =1. Then G = Aut(A6), p = 2 and (A, B) G {(S, S), (minG(S, S), S), (S, minG(S, S))}.

Lemma 2.3 ( [1]). Let G > Gi > G2, Gi > A, G > B. Suppose that_G2 n Bh = 1 for some h G G and in the quotient group Gi = Gi/G2 we have A n (Gi n Bh)f = 1 for some f G Gi. Then A n Bg = 1 for some g G G.

In conclusion of this part we note a simple example the group G with subgroups A and B for which MinG(A, B) = minG (A, B).

Let G be the symmetric group on the four symbols and S G Syl2(G). Then S ~ D8 and O2(G) ~ Z2 x Z2. Take the subgroup S as A and as B we take the subgroup of order four of S, which not belongs to O2(G). Then Bg n O^G^ = 2 for any g G G. Since A n B = B and A n Bf | = 2 for Bf ^ A, then M = {B, Bf n O2(G),Bf2 n O2(G)}, where f | = 3. Therefore MinG(A, B) = S = O2(G) = minG(A, B).

3. Some properties of the group Aut(F4(q))

Further, $ is a reduced indecomposable root system, n = {r1, ...,ri} is its set of fundamental roots, $+ is the positive root system respect to n, and also = —$+.

Denote by $(q) an adjoint Chevalley group of type $ of rank l over the finite field Fq of the order q = pn, where p is a prime. The group $(q) is generated by the root subgroups Xr = (xr (t) | t e Fq), r e $, where xr (t) is the corresponding root element in the group $(q). We will need the following natural subgroups of the group $(q): the unipotent subgroups U = (Xr I r e $+), V = (Xr I r e the monomial subgroup N = (nr(t) | r e $, t e F*), the diagonal subgroup H = (hr(t) | r e $, t e F*) and the Borel subgroup B = UH. Here, F* is the multiplicative subgroup of the field Fq and nr(t) = xr(t)x-r(—t-1)xr(t), hr(t) = nr(t)nr( —1). We set also I = {1,2,...,l}.

Overgroups of the Borel subgroup B and conjugate with them are called parabolic. Due to familiar result of J.Tits, parabolic subgroups containing subgroup B are PJ = (B,nrj(1) | j e J), where J C I.

Lemma 3.1 ( [5], Lemma 5). Fix a monomial element n0 with condition Un° = V and a positive integer i e I. Set n = n0nr. (1). Then U n Un = Xri.

For l = 1 the root subgroup Xri coincides with a Sylow p-subgroup of the group $(q) and in this case in the Lemma 3.1 the element n is diagonal.

Further, n = {r1,r2,r3,r4} is a fundamental root system of type F4, moreover r1, r2 are short roots and r2 + r3 is a root. The graph automorphism t of the Chevalley groups F4(2) is defined correctly by symmetry of order 2 of the Coxeter graph of type F4, which induces the bijection r ^ r of the root system of type F4 to itself such that —r = —r [6, Lemma 12.3.2]. Note, by the way, that root system of type F4 is the union subsystems $1 and $2 of type B4 and C4 respectively and $1 = $2 (see, for Example, [7]).

Lemma 3.2. Let U, V be the unipotent subgroups of the group F4(2) with the graph automorphism t of order 2 as above. Then S = U X (t) is a Sylow 2-subgroup of the group Aut(F4(2)) = F4(2)(t) and there is an unique monomial element n0 such that S n Sn0 = (t).

Proof. Just the last equality requires justification. In the group F4(2) there is an unique monomial element n0 such that Un0 = V. Since Xn0 = X-r, XT = Xr and —r = —r, then n0t=Tn0. Hence, S n Sno = (t). □

Lemma 3.3. Let S be as in Lemma 3.2. Then in the group F4(2) there is a monomial element n such that S n Sn = Xri.

Proof. Let a monomial element n0 be as in Lemma 3.1 and n = n0nri (1). By Lemma 3.1 we have U n Un = Xri. Now using the equalities n0T=Tn0 (see proof of Lemma 3.2) and nri (1)Tnri (1) = nri (1)nr4 (1)t we obtain the assertion of lemma S n Sn = Xri. □

Lemma 3.4. Let i = 1 or 4 and P = Pi\{i} be a maximal parabolic subgroup of the group F4(q). Then (XU) = Op(P).

Proof. Let P = PI\{4}. Then we have equality

Op(P) = (Xr | r = ckrk + • • • + C4r4, 1 < k < 4, cj > 1).

Further, for a root r = ar1 + br2 + cr3 + dr4 we will use the notation abcd. Using this compact representation of roots and the table VIII for the root system of type F4 in [8], we have equality

Op(P) = (Xr | r e *),

where

V = {0001,0011, 0111,1111, 0211,0221,1211,1221, 2211,1321, 2221, 2321, 2421, 2431, 2432}. Evidently, Xoooi Q M. For any t,u e Fg the commutator formula of Chevalley gives the equalities

[xoooi(i),xooio(w [xoooi(t),xo2io(u [xo2ii(t),xooio(u [xooii(t), xoioo(u [xoiii(t),xiooo(u [xiiii(t),xoioo(u [xo2ii(t),xiooo(u

= xooii(±tu), = xo2ii(±tu), = xo22i(±tu), = xoiii(±tu)xo2ii(±tu2), = xiiii(±tu), = xi2ii(±tu), = xi2ii(±tu)x22ii(±tu2),

[x22ii(t),xooio(u)] = x222i(±tu), [xo22i(t),xiooo(u)] = xi22i(±tu)x222i(±tu2), [xi2ii(t),xooio(u)] = xi22i(±tu)x2432(±t2u), [xi22i(t),xoioo (u)] = xi32i (±tu), [xi32i(t),xiooo(u)] = x232i(±tu), [x22ii(t),xo2io(u)} = x242i(±tu), [x222i(t),xo2io(u)} = x243i (±tu).

Using these equalities, we successively obtain the inclusions Xr Ç Op(P) for all r G The

conclusion of the lemma is also true for i = 1 by the equality PT\{4y = Pi\{i}

4. Some properties of Sylow p-subgroups of the groups of Lie type over fields of characteristic p

Analogues of the subgroups Xr, U, V, N, H, B, Pj of the Chevalley group &(q) in Section 3 are also defined for twisted Chevalley group n§(q). In this section, G(q) is a group of Lie type over a finite field of order q of characteristic p, where G = $ or n$. It is well known that any parabolic subgroup PJ of the group G(q) is a semidirect product with kernel Op(PJ) and a noninvariant factor L. A subgroup L is called a Levi factor and it is isomorphic to the central product of groups of Lie type of smaller ranks over the initial field.

We will need the following strengthening of Lemma 3.13 from [3].

Lemma 4.1. The number of orbits under the action of conjugation by elements of U on the set of subgroups Ug of G(q) with the condition U n Ug = 1, g G G(q), is equal to one. Moreover, the length of this single orbit is \U| and it consists of subgroups of the form Vu, u G U.

Proof. Any element g G G(q) can be uniquely represented in the form g = unwv, where u,v G U, nw G N, and nw vn-i G V. Let U n Ug = 1. Then U n Un™ = 1. Since X?™ = Xw(r), then w($+) = —. Thus, any subgroup Ug with the condition U n Ug = 1 has the form Vu for some u G U. Since NG(q)(V) = HV, then the number of subgroups of the form Vu, u G U, with the condition U n Vu = 1 is equal to \U |. □

Lemma 4.2. V n Op(PJ) = 1 and the subgroup V covers the Sylow p-subgroup in the quotient group Pj = Pj/Op(Pj).

Proof. Since Op(PJ) C U, and U n V = 1, then Op(PJ) n V = 1. By virtue of the Levi decomposition ^ylp(Pj^)| = ^nVl Consequently, the subgroup V covers the Sylow p-subgroup of the quotient group PJ = PJ/Op(PJ). □

5. The proof of the Theorem 1

Further in the proof, we use the notations of the Section 3 for subgroups and elements of the group Aut(F4(2)).

So, by the hypothesis of the theorem, G is a finite group, Soc(G) ~ F4(2), S e Syl2(G) and MinG(S,S) = 1. Since Aut(F4(2)) = F4(2)(t) then it is possible only two cases: 1) G = F4(2); 2) G = F4 (2) (t).

The first case is not possible, because the Sylow 2-subgroups U and V of the group G = F4(2) have the unite intersection.

Let G = F4 (2) (t). Without loss of generality, we can assume that S = U(t). Let g e G. Then g = u1nu20, where u1,u2 e U, n e N and 0 e (t) . If n = n0 then U n Ug = 1 and, consequently , S n Sg = 1. If n = n0, then we have S n Sg = (t) by Lemma 3.2. Thus S n Sg = 1 for each g e G and moreover any element (subgroup) of the set m for A = B = S has order 2. Set

P = P{2,3}.

By Lemma 3.1 there is a monomial element n e F4(2) such that S n Sn = Xri. By Lemma 3.4

I\{i})

(XU) = O2(Pi\

for i =1 or 4. Since (O2(PAW), O2(Pj\{i})) = O2(P), then O2(P) < minG(S,S). Let

N = Ng(p ) = P (t ).

Then O2(N) = O2 (P) and

N = N/O2(N) ~ Aut(A6). We choose an element x e G such that the intersection of cardinality 2

D = S n Sx e m

does not lie in O2(N). (Such an element certainly exists, for example, as x, we can take the element n0 from Lemma 3.1.) Since O2(N) C S, then O2(N) n Sx = 1. Set

S1 = N n Sx.

By Lemma 4.3, the subgroup Ux, and therefore by definition, the subgroup

u1 = p n u x < s1

covers a Sylow 2-subgroup of the factor group

P = P/O2 (P) ~ Sp4(2).

Socle of the group Sp4(2) is isomorphic to A6, but Sp4(2) ^ Aut(A6). Therefor min^(U, U1) = 1 by Lemma 2.1. Hence, also min^(U, U1) = 1, since N : P| = 2 and U, U1 < P.

We show that S1 = U1. Suppose the contrary, let S1 = U1. Then in the quotient group N we have

min^(S, S1) = min^(S, U1) = min^(U, U1) = 1.

Moreover, to obtain the second equality, we also use the fact that U1 covers a Sylow 2-subgroup of P and \N : P| = 2. Now, by Lemma 2.3, with G = G, G1 = N, G2 = O2(N) and A = B = S, by O2(N) n Sx = 1, we have S n Sy = 1 for some y e G. That is, minG(S, S) = 1. A contradiction.

So, S1 = U1. Therefore, and by N : P| =2, the subgroup S1 covers a Sylow 2-subgroup of N. Since O2(N) < minG(S, S), to describe the subgroup minG(S,S) it is necessary to know its image minG(S, S) in N = N/O2(N). We show that minG(S, S) = min^(S, S).

Suppose that D = S n Sx e m does not lie in the preimage of S2 in the N of the subgroup min^(S, S), which by virtue of Lemma 2.1 is isomorphic to the dihedral group of order 16. Then S2 n S1 = 1, since S2 < S, and S1 < Sx. From here min^(S, S) n S1 = 1. But this is impossible, because S1 e Syl2(N). Thus, D < S2 and, consequently, minG(S,S) < min^(S, S).

On the other hand, each element D e m is of order 2 by Lemma 2.1 and by definition D = S n S1 for some y e N. Therefore, for the preimage D ^ S of the subgroup D we have D : O2(N)| = 2. Hence, D < SfO2(N) and D ^ O2(N), otherwise S n Sf = I. Therefore, DnSf | = 2. Further, DnSv = Dn(NnSx)V = DnSxy. We show that DnSxy = SnSxy. Indeed, from D n Sf | =2, we obtain D = (d)O2(N), where d is an involution. Since Sxy n O2(N) = 1, then the image in S of the intersection D1 = S n Sxy contains isomorphic to D1 copy D1. Obviously, D1 is also contained in the intersection N n Sxy = (N n Sx)V = Sf, and subgroup Sf is isomorphic to its image S1 e Syl2(N). Therefore, D1 ~ D1 < D. Since D = 2, then [D^ = 2 = = S n Sxv l Hence D n Sxy = S n Sxy e m. So, we have the correspondence D ^ (d) = D n Sf = S n Sxv e m. Therefore the subgroup minG(S, S) covers the subgroup min^(S, S). Hence, min^(S,S) ^ minG(S, S).

Thus, we have established that minG(S, S) = min^(S,S). Now Theorem 1 follows from Lemma 2.1.

Theorem 1 is proved.

6. The proof of the Theorem 2

So, by the hypothesis of the theorem, G is a finite group, Soc(G) ~ F4(2), A, B are primary p-subgroup of G, and S is a Sylow 2-subgroup of G.

(1) ^ (2). Let MinG(A, B) = 1. Then also minG(A, B) = 1. In view of Theorem B(2) of [3], G ~ Aut(F4(2)) and the subgroups A and B are 2-groups. Without loss of generality we can assume that A and B lie in S. Let the set m corresponds to the subgroup minG (S, S) = 1. As shown in the proof of Theorem 1, all elements of the set m have order 2. Therefore, if an element of the m is not in minG(A, B), then minG(A, B) = 1, but this is impossible by assumption. Hence, minG(S,S) < minG(A,B). Since A and B are 2-groups and |S : minG(S,S)| =2 by Theorem 1, then the subgroups A and B coincide with the subgroups S or minG(S, S).

We show that the pair (A,B) = (minG(S,S), minG(S, S)) is excluded. Again, in view of Theorem 1

minG(S,S) = O2(P{2,s}) ■ minL{T} (S^) < O2(P{2,s}) X L(t),

where S1 is a Sylow 2-subgroup of the group L(t) ^ Aut(A6), and minL^T^(S1,S1) ^ D16. By Lemma 3.2, S n Sn0 = (t), therefore, minG(S, S) n (minG(S, S))n0 < (t). In particular, O2(P{2,3}) n O2(P{2t3})n° = 1. Since n0T = Tn0, then (L(t))n0 = L(t). Summarizing all of the above and applying Lemma 2.2, we obtain the existence of an element g e L(t) such that minG(S, S) n (minG(S, S))g = 1.

(2) ^ (1). If (A, B) = (S, S), then MinG(A, B) = 1 in view of Theorem 1.

Let (A,B) = (minG(S, S),S). Suppose that MinG(A, B) = 1. Then also minG(A, B) = 1. Therefore, S n (minG(S,S))f = 1 for some y e G and S n Sv| = 1 since minG(S,S) = 1. As noted above, S1 : minG(S, S)| = 2 by Theorem 1. Moreover, it follows from Theorem 1 that S = minG(S, S) X (i) for any involution i e S\minG(S, S). Therefore, |S n Sv| = 2, otherwise S n (minG(S, S))v = 1. Thus, S n Sv e m and S n Sv < minG(Sf, Sv) = (minG(s, S))v. This is a contradiction.

The case (A, B) = (S, minG(S, S)) is considered similar to the case (A, B) = (minG(S, S),S). Theorem 2 is proved.

The first author was supported by the RNF (project 15-11-10025), Theorem 1, as well as agreements between the Russian Federation Ministry of Education and Science and Ural Federal University on 08/27/2013, number 02.A03.21.0006, Theorem 2. The work of the second author was supported by the RFBR (project 16-01-00707).

References

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[4] V.I.Zenkov, On intersection of nilpotent subgroups in finite groups with socle L2(q), Siberian Math. J., 57(2016), no. 6, 1280-1290 (in Russian).

[5] V.I.Zenkov, Ya.N.Nuzhin, On intersection of primary subgroups of odd order in finite almost simple groups, Fundamental. i prikladnaya matematika, 19(2014), no. 6, 115-123 (in Russian).

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О пересечениях примарных подгрупп в группе Aut(F4(2))

Виктор И. Зенков

Институт математики и механики УрО РАН Ковалевской, 16, Екатеринбург, 620990

Россия

Яков Н. Нужин

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Показано, что в конечной группе G, изоморфной группе всех автоморфизмов группы Шевалле F4(2), существуют лишь три типа упорядоченных пар примарных подгрупп A и B с условием: A П Bg = 1 для любого g £ G. Приведено описание всех упорядоченных пар (A, B) таких подгрупп с точностью до сопряженности в группе G, в частности, доказано, что A и B являются 2-группами.

Ключевые слова: конечная группа, почти простая группа, примарная подгруппа.