Centralizers of finite p-subgroups in simple locally finite groups Текст научной статьи по специальности «Математика»

УДК 512

Centralizers of Finite p-Subgroups in Simple Locally Finite Groups

Mahmut Kuzucuoglu*

Department of Mathematics Middle East Technical University Ankara, 06531

Turkey

Received 26.10.2016, received in revised form 06.12.2016, accepted 08.03.2017 We are interested in the following questions of B. Hartley: (1) Is it true that, in an infinite, simple locally finite group, if the centralizer of a finite subgroup is linear, then G is linear? (2) For a finite subgroup F of a non-linear simple locally finite group is the order \CG(F)| infinite? We prove the following: Let G be a non-linear simple locally finite group which has a Kegel sequence K = {(Gi, 1) : i £ N} consisting of finite simple subgroups. Let p be a fixed prime and s £ N. Then for any finite p—subgroup F of G, the centralizer Cg(F) contains subgroups isomorphic to the homomorphic images of SL(s, Fq). In particular Cg(F) is a non-linear group. We also show that if F is a finite p-subgroup of the infinite locally finite simple group G of classical type and given s £ N and the rank of G is sufficiently large with respect to \F\ and s, then Cg(F) contains subgroups which are isomorphic to homomorphic images of SL(s,K).

Keywords: centralizer, simple locally finite, non-linear group. DOI: 10.17516/1997-1397-2017-10-3-281-286.

Hartley asked the following question: Let G be a simple locally finite group containing a finite subgroup with linear centralizer. Does it follow that G is linear? He proved in [2, Theorem A] the following: Let G be any non-linear simple locally finite group and F be a finite subgroup of G. Then there exist subgroups D < C < G such that D contains [C, F] and C n F, and C/D is a direct product of finite alternating groups of unbounded orders. So the above question is not answered positively in the general case because [C, F] is not necessarily identity.

By standard methods, one may reduce the question to countable non-linear simple locally finite groups. So, we may assume that G is a non-linear countable, simple locally finite group. In [6] it is mentioned that, the structure of centralizers of simple locally finite groups which has a Kegel sequence as a union of finite simple subgroups and the ones which has no, particular type of such Kegel sequences are quite different.

Recall that an element in a simple group of Lie type is semisimple if its order and the characteristic of the field is relatively prime. In the alternating groups all elements are semisimple.

Definition 1. A subgroup F of a finite non-abelian simple group G is called a totally semisimple subgroup if every element of F is a semisimple element in G whenever it is a simple group of Lie type. If G is alternating, then all finite subgroups are totally semisimple.

Observe that, a simple locally finite group G has a local system consisting of finite simple subgroups if and only if G has a Kegel sequence K = {(Gi, 1) | i e N}. For more information about Kegel sequences see [3]. Our main result is the following.

Theorem 2. Let G be a non-linear simple locally finite group which has a Kegel sequence K = {(Gi, 1) : i e N} consisting of finite simple subgroups. Let p be a fixed prime and s e N. Then

* matmah@metu.edu.tr © Siberian Federal University. All rights reserved

for any finite p—subgroup F of G, the centralizer Cq(F) contains subgroups isomorphic to the homomorphic images of SL(s, Fg). In particular Cq(F) is a non-linear group.

Proof. Step 1. Let F be a finite p-subgroup for a fixed prime p of the simple locally finite non-linear group G which has a Kegel sequence consisting of finite simple subgroups Gi. By the classification of the finite simple groups, we may assume that G has a Kegel sequence for a fixed type of classical group. Let pi be the characteristic of the field over which the group Gi is defined. Let p = {pi | i € N} be the set of all these primes. If the set p is infinite, then we may pass to a subsequence such that each Gi is defined over a field of characteristic pi and pi = pi+\. In this case, by deleting the fixed prime p from the infinite list and if necessary by passing to a subsequence, we may assume that our p does not belong to the set of primes which appear as a characteristic. Hence F becomes a totally semisimple subgroup in each Gi. Otherwise the set p is finite. We may pass to a subsequence such that each Gi is a simple group over a field of fixed prime r. If p = r then again F becomes a totally semisimple subgroup in each Gi.

By the above paragraph we may choose a subsequence such that either F is a totally semisimple subgroup in each finite simple group Gi of fixed classical type for all i € N and so theorem is proved in the totally semisimple and alternating cases in [1], or F is a p-group in Gi of fixed classical type defined over a field of characteristic p.

So we are in the case that F is a p-group in Gi where Gi is of fixed classical type defined over a field of characteristic p.

Reduction of centralizer from projective special classical groups to classical groups case.

As Cq(F)N/N < Cq/n(F/N) if Cq(F)N/N contains subgroups isomorphic to SL(s, Fg), then so is Cq/n (F/N).

By using this we may reduce the proof of the Theorem to Classical groups.

Step 2. Let p be a fixed prime. Let F\ be a finite p-subgroup of the finite simple group G of classical type defined over a field of characteristic p. G is constructed from a vector space V of dimension m and G = T/Z(T) where T = SL(V), Sp(V), Q±(V) or SU(V). Since (IZ(T)|, p) = 1, by Schur-Zassenhaus theorem, the inverse image L of F\ in T can be written as L = F x Z(T) where F is a finite p-subgroup of T isomorphic to F\. Then CT(L) = CT(F) and

Ct(F x Z(T))/Z(T) = Ct(F)Z(T)/Z(T) < Ct/z(t)F) = Cq(F{) moreover as CT(F) > Z(T), the order

I(Ct(F) n Z(T))| = IZ(T)| = (m, |F,| — 1) < |F,| = q

So in order to prove the Theorem, it is enough to show that, for finite p-subgroup F of T the group CT(F) contains subgroups isomorphic to homomorphic images of SL(s, Fg). First we show this for SL(n, Fg).

Lemma 3. Let F be a finite p-subgroup of SL(n, Fg) or GL(n, Fg) where Fg be a finite field of order q of characteristic p. Then CSL(n,Fq) (F) has subgroups isomorphic to SL(n, Fg) provided that n is large.

Proof. Let Vn be an n-dimensional vector space over Fg on which F acts. First we show n

dim(Cy (F)) > ——. We may prove this by induction on Fl |F|

Assume that F| = p. Since characteristic of the field is p and the element x € F is of order p, we have xp = 1, which implies (x — 1)p = 0. Then F has either Jordan block of size p or it fixes the given vector. As we wish to show that F fixes a subspace of large dimension, we may assume that, F has Jordan blocks of size p on the whole space. In this case Vn can be written as a direct sum of the corresponding F invariant subspaces and in each F invariant subspace

we have an eigenvector corresponding to the eigenvalue 1 and hence it is fixed. Then we have n n

dim(CV (F)) > — = —— > number of Jordan blocks of F on Vn. p | F|

Now assume that |F| > p. Since F is a p-group, there exists a normal subgroup H < F such

n

that IF : H| = p. By induction assumption dim(CV (H)) > ——. Now consider Cv (F). The

| H|

group F acts on CVn (H) = W as a cyclic group of order p as H acts trivially on W. Hence

dim(W) n dim(CVn(F)) > -= by the first paragraph.

We may form a basis for Vn extending the basis of CVn (F). Consider the non-singular linear

transformations of CVn (F) which acts trivially on the remaining basis vectors of Vn. Then

n

Cvn(F) has a subgroup isomorphic to GL(jfj, Fg) and contained in CGL(n,Fq)(F). □

Now we prove the above result for the other classical groups that will complete the proof of the Theorem.

Observe that in the next Lemma, classical groups over fields of characteristic 2 is also included, main reference is [2]. In particular if G is an orthogonal group over a field of characteristic 2, then we assume dim(Vm) is even as in [2].

Lemma 4. Let G = G(m, Fg) be unitary, symplectic or orthogonal group over a finite field Fg of characteristic p. Let F be a finite p-subgroup of G. Let s be a given integer. If m ^ 4|F| + 2s|F^ then Cg(F) contains subgroups isomorphic to SL(s, Fg).

Proof. Let Vm be an m dimensional vector space over a finite field Fg of characteristic p associated to the group G and ( , ) be a non-degenerate symmetric, unitary or symplectic form on Vm. For the orthogonal groups over a field of characteristic 2, the vector space Vm is associated with a quadratic form g : Vm ^ Fg together with a Fg valued bilinear form ( , ) on Vm such that g(Xx + fj,y) = X2g(x) + X^(x, y) + n2g(y) where x,y e Vm and X, ^ e Fg.

By [2, (b), p. 508] if m > 2s|F| +4|F|, then F leaves invariant a totally isotropic (respectively totally singular) subspace of Vm of dimension at least sFl Then F acts on this F invariant subspace of dimension sF| and by Lemma 3, if dimension of the totally isotropic subspace > sF|, then CSL(m, Fq)(F) contains a subgroup isomorphic to SL(s, Fg). Then by Witt extension theorem we may extend the action to the isometries of the vector space Vm and hence CG(F) contains subgroups isomorphic to SL(s, Fg). □

Completion of the Proof of Theorem 2. Let s be a given integer. In a non-linear locally finite simple group with a given finite subgroup F, we may find a classical group Gi where the rank of Gi is sufficiently large. Then by Lemma 3 and Lemma 4, CGi (F) contains a subgroup which is isomorphic to SL(s, Fg). Hence CG(F) contains subgroups isomorphic to homomorphic images SL(s, Fg) for any s e N. In particular CG(F) is a non-linear group. □

Theorem 5. If F is a finite p-subgroup of the infinite locally finite simple group G of classical type and the rank of G is sufficiently large with respect to |F^ then CG(F) contains subgroup isomorphic to homomorphic images of SL(s, K) where K is a locally finite field. In particular CG(F) is an infinite group.

Proof. If F is a totaly semisimple element of G, then the result can be extracted from [1, Theorem 1.11] and the proof of [1, Theorem 2.1]. If F is a p-subgroup of a locally finite simple group defined over a field of characteristic p, then Lemma 3 and Lemma 4 give the result. □

Definition 6. Let K = {(Gi: Ni) | i e I} be a Kegel cover of a simple locally finite group G. A finite p-subgroup F of G is called a K-p-subgroup if (p, N^) = 1 for all i e I.

In general, for any non-linear, simple locally finite group, it is not true that, every finite p-subgroup is a K-p-subgroup. In [7] Meierfrankenfeld showed that, there exists a simple non-linear locally finite group G such that CQ(x) is a p-group for an element x of order p. But by [1], if G has a Kegel sequence as above, then CQ(x) involves an infinite non-linear simple subgroup. So the groups in [7] do not have such a nice Kegel sequence. Our results can be used in this direction to decide whether G has a nice Kegel sequence or not; provided that we know the structure of the centralizers of its elements.

Corollary 7. Let G be a non-linear simple locally finite group and K = {(Gi,Ni) | i € N} be a Kegel sequence of G. Then for any finite K-p-subgroup F, the centralizer CQ(F) contains subgroups isomorphic to the homomorphic images of SL(s, K). In particular CQ(F) is an infinite group.

Proof. This result is an easy application of splitting of the centralizer

CQi/Ni (FNi/Ni) = CQ (F )Ni/Ni.

□

Lemma 8. Let G be an infinite simple linear group over a locally finite field K of characteristic p and F be a p-subgroup of G (F could be infinite). Then ^Q(F)| ^ K|. Moreover if K| is infinite, then CQ(F)| is infinite.

Proof. By Zorn's lemma every p-subgroup of G is contained in a maximal p-subgroup P of G. As maximal p-subgroups of G are nilpotent; P is nilpotent and hence Z(P) < CQ (F). As Z(P) contains a subgroup isomorphic to the additive group of K we have whenever |K| is infinite, then Cq(F)| is infinite. □

Definition 9 ( [1]). Let G be a simple linear algebraic group. A finite abelian subgroup A consisting of semisimple elements in G is called a d-abelian subgroup if one of the following holds:

1. The root system associated with G has type A¡, and the Hall n-subgroup of A is cyclic where n is the set of primes dividing l + 1.

2. The root system associated with G has type B¡, C¡,D¡ or G2 and the Sylow 2-subgroup of A is cyclic.

3. The root system associated with G has type E6, E7 or F4 and the Hall-{2, 3}-subgroup of A is cyclic.

4. The root system associated with G has type E8 and the Hall-{2, 3, 5}-subgroup of A is cyclic.

Theorem 10. Let G be an infinite simple classical group of rank l over a field of characteristic p and F be a finite subgroup of G with F = P x Q where P is a p-subgroup and Q is p'-part of F. Let s be a given integer. If Q is d-abelian and l is sufficiently large with respect to |F| and s, then Cq(F) contains subgroup isomorphic to homomorphic images of SL(s, K). In particular Cq(F) is an infinite group.

Proof. Let G be an infinite simple locally finite group over a field K of characteristic p. Let G be the simple linear algebraic group of adjoint type over an algebraically closed field K which we obtain G as a union of fixed points of Frobenius automorphisms an where ni ^.1+1, i € N, for details see [1]. Then Q is a d-abelian subgroup of G and by [8, Theorem 5.8(c) and Exercise 5.11], Q is contained in a maximal torus fixed by ani for all i. The group Cq(Q)° is generated by T and the Ua with a € Si where Si be the system of roots vanishing on Q and Cq(Q)° is reductive group by [8, 4.1 (b) E35] with Si as its root system.

Since the order of Cq(Q)/Cq(Q)° is a finite fixed number it is enough to consider the theorem for Cq(Q)°.

By [4, p 20, Proposition ] if s G G is semisimple, then s G CG(s)° and every unipotent element of Co(s) lies in Cq(s)° hence the p-part P of F also lies in Cq(Q)° . The group (Cq(Q)°) is a semisimple subgroup and P is a finite p-subgroup which is fixed by the Frobenius automorphisms ani. The automorphisms ani are automorphisms of CG(Q)° see [8, 3.2, E.10.] and so automorphisms of (Cq(Q)°) . Our group P lies in the semisimple part of Cq(Q)°. Let (Cq(Q)°)' = H\H2.. .Hk be the product of simple algebraic groups Hi. Then the Frobenius automorphism acts on the components and choose an orbit of ani containing a component of large rank. Since the rank of Cq(Q)° is sufficiently large and this rank is the sum of the ranks of the simple components Hi, we have such a component. For the existence of this large rank see [5]. Then the rank of CCo(Q)0 (P) is sufficiently large. Then the fixed points of the automorphisms ani on this component gives the centralizers of large cardinality as the fixed points contains the classical groups of large cardinalities and certainly they are in the centralizer of F.

Then the fixed points of the automorphisms on this component gives the centralizers which proves the theorem. □

Dedicated to V. M. Levchuk and A. Yu. Ol'shanskii on the occasion of their 70th birthday.

References

[1] K. Ersoy, M. Kuzucuoglu, Centralizers of subgroups in simple locally finite groups, J. Group Theory, 15(2012), no. 1, 9-22.

[2] B. Hartley, Centralizing Properties in Simple Locally Finite Groups and Large Finite Classical Groups, J. Austral. Math. Soc. (Series A), 49(1990), 502-513.

[3] B. Hartley, M. Kuzucuoglu, Centralizers of elements in locally finite simple groups, Proc. London Math. Soc., 62 (1991), no. 3, 301-324.

[4] J. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, AMS, Mathematical Surveys and Monographs, 1995.

[5] M. Kuzucuoglu, Centralizers of semisimple subgroups in locally finite simple groups, Rend. Sem. Mat. Univ. Padova, 92(1994), 79-90.

[6] M. Kuzucuoglu, Centralizers in simple locally finite groups, International Journal of Group Theor., 02(2012), no. 1, 1-10.

[7] U. Meierfrankenfeld, Locally Finite Simple Group with a p-group as centralizer, Turkish J. Math., 31(2007), 95-103.

[8] T.A.Springer, R.Steinberg, Conjugacy Classes in Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes in Math., 131, 1970, Springer-Verlag, Berlin.

Централизаторы конечных р-подгрупп в простых локально конечных группах

Махмут Кузусуоглу

Департамент математики Средневосточный технический университет

Анкара, 06531

Турция

Нас интересуют следующие вопросы Б.Хартли: (1) Правда ли, что в бесконечной простой локально конечной группе, если централизатор конечной подгруппы линейный, то G является линейной? (2) Для конечной подгруппы F нелинейной простой локально конечной группы порядок |CG(F)| бесконечен?

Доказывается следующее: пусть G — нелинейная простая локально конечная группа, имеющая последовательность Кегеля K = {(Gi, 1) : I G N}, состоящую из конечных простых подгрупп. Пусть p — фиксированное простое число, s G N. Тогда для любой конечной р-подгруппы F группы G централизатор Cg(F) содержит подгруппы, изоморфные гомоморфному образу SL(s, Fq). В частности, CG(F) является нелинейной группой. Мы также показываем, что если F — конечная р-подгруппа бесконечной локально конечной простой группы G задачи классического типа и заданных s G N, и ранг G достаточно большой относительно IF| и s, то CG(F) содержит подгруппы, изоморфные гомоморфным образам SL(s,K).

Ключевые слова: централизатор, простая локально конечная, нелинейная группа.