In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q

Ersoy, Kıvanç

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Glasgow Math. J. 62 (2020) 183–186. C© Glasgow Mathematical Journal Trust 2019. This is an Open Access article, distributedunder the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permitsunrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.doi:10.1017/S001708951900003X.CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLYFINITE GROUPSKIVANÇ ERSOYInstitute of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany andDepartment of Mathematics, Mimar Sinan Fine Arts University, Istanbul 34427, Turkeye-mail: ersoykivanc@gmail.com(Received 17 May 2018; revised 17 December 2018; accepted 7 January 2019;first published online 21 February 2019)Abstract. In Ersoy et al. [J. Algebra 481 (2017), 1–11], we have proved that if G is alocally finite group with an elementary abelian p-subgroup A of order strictly greater thanp2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α)does not involve an infinite simple group, then G is almost locally soluble. This result is aconsequence of another result proved in Ersoy et al. [J. Algebra 481 (2017), 1–11], namely:if G is a simple locally finite group with an elementary abelian group A of automorphismsacting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikovand for every non-identity α ∈ A, the set of fixed points CG(α) does not involve an infinitesimple groups then G is finite. In this paper, we improve this result about simple locallyfinite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) isChernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involvean infinite simple group. We prove that if Aut(G) has such a subgroup, then G ∼= PSLp(k)where char k = p and P has a subgroup Q of order p2 such that CG(P) = Q.2010 Mathematics Subject Classification. Primary 20E32, 20E36; Secondary 20G151. Introduction. In [2], we have proved the following result:THEOREM 1.1. [2, Theorem 1.1]. Let p be a prime and G a locally finite group contain-ing an elementary abelian p-subgroup A of rank at least 3 such that CG(A) is Chernikov andCG(a) involves no infinite simple groups for any a ∈ A#. Then G is almost locally soluble.To prove Theorem 1.1, we gave the following characterization of PSLp(k) wherechark = p.THEOREM 1.2. [2, Theorem 1.2]. An infinite simple locally finite group G admits anelementary abelian p-group of automorphisms A such that CG(A) is Chernikov and CG(a)involves no infinite simple groups for any a ∈ A# if and only if G is isomorphic to PSLp(k)for some locally finite field k of characteristic different from p and A has order p2.In this paper, we will improve Theorem 1.2. Indeed, we will prove a similar resultwithout assuming A is an elementary abelian, but instead, we prove for any subgroup ofexponent p.THEOREM 1.3. Let G be an infinite simple locally finite group, P a subgroup ofautomorphisms of exponent p such that(1) CG(P) is Chernikov,available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S001708951900003XDownloaded from https://www.cambridge.org/core. Charité Universitätsmedizin Berlin, on 17 May 2021 at 11:47:30, subject to the Cambridge Core terms of use,184 KIVANÇ ERSOY(2) For every α ∈ P\{1}, the set of fixed points CG(α) does not involve an infinitesimple group.Then G ∼= PSLp(k) where k is an infinite locally finite field of characteristic p and P has asubgroup Q of order p2 such that CG(P) = CG(Q) = Q.2. Preliminaries. Let us recall some definitions of the concepts mentioned in thetheorems. First, consider Cpn = {x ∈ C : xpn = 1}. Here, (Cpn , .) defines a group isomorphicto a cyclic group of order pn. Observe that if m|n then Cpm ≤ Cpn , and with the inclusionmaps, these sets form a direct system, where the direct limitlimn∈NCpnis denoted by Cp∞ , which consists of all complex pn-th roots of unity, and forms a groupunder complex multiplication. This group is called the quasi-cylic p-group.DEFINITION 2.1. A group is called a Chernikov group if it is a finite extension of adirect product of finitely many copies of some quasi-cyclic pi-groups, for possibly distinctprimes pi.DEFINITION 2.2. Let χ be a group-theoretical property. If a group G has a normalsubgroup of finite index satisfying χ , then G is called almost χ .DEFINITION 2.3. Let G and H be two groups. If G has a normal subgroup K such thatG/K has a subgroup isomorphic to H, then G is said to involve a subgroup isomorphicto H .DEFINITION 2.4. A group satisfies the minimal condition, namely min, if anynon-empty set of subgroups has a minimal subgroup. A group satisfies min-p if anynon-empty set of p-subgroups has a minimal subgroup.Kegel–Wehrfritz and Sunkov proved independently that a locally finite group satisfy-ing minimal condition is a Chernikov group (see [5, 9]). For detailed discussion of groupssatisfying min and min-p, see [6].3. Main results. First, we need the following proposition:PROPOSITION 3.1. Let G be a simple linear algebraic group of adjoint type over thealgebraic closure of Fq, let g ∈ G be an element of prime order p = q such that CG(g) is anon-abelian group which does not involve any infinite simple groups. Then(i) The identity component CG(g)0 of the centralizer of g in G is a maximal torus of G,(ii) G ∼= PGLp(Fq).Proof. Since G is a simple linear algebraic group of adjoint type over the algebraicclosure of Fq and g ∈ G a semisimple element, g is contained in a maximal torus T of G.By [7, Propositions 14.1 and 14.2], CG(g)0 is connected reductive, containing a maximaltorus T , and involving no infinite simple groups. Hence, CG(g)0 = T . By [7, Proposition14.20], the exponent of CG(g)/CG(g)0 divides p, hence either CG(g) is connected, andhence a torus, or CG(g)/CG(g)0 is a finite group of exponent p.Since CG(g) is not abelian, one has CG(g) a finite extension of an abelian group T , soit has finite rank. Recall that an infinite group G is said to have finite rank r if every finitelyavailable at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S001708951900003XDownloaded from https://www.cambridge.org/core. Charité Universitätsmedizin Berlin, on 17 May 2021 at 11:47:30, subject to the Cambridge Core terms of use,CENTRALIZERS OF p-SUBGROUPS 185generated subgroup is r-generated. In [1, Theorem 1.8], we have shown that when a simplelinear algebraic group G over the algebraic closure of Fq has an element g of order p withCG(z) has finite rank, then one of the following cases occur:(1) G is of type Al and p > l,(2) G is of type Bl, Cl, and p > 2l − 1,(3) G has type Dl and p > 2l − 3,(4) G is isomorphic to one of E6, E7, E8, F4, or G2 and p > 11, 17, 29, 17, or 5,respectively.On the other hand, since CG(g)/CG(g)0 has exponent p, by [10, Corollary 4.4] and [7,Proposition 14.20], we get p is a torsion prime. The list of torsion primes of linear algebraicgroups is defined as follows: for type Al, these are the primes that divide l + 1. For typesBl, Cl, Dl, G2, the prime is 2. For types E6, E7, F4, the primes are 2 and 3, and for type E8,the primes are 2, 3, 5 (see [8]).Hence, one deduce that the only possible case that may occur is G has type Ap−1,indeed G ∼= PGLp(Fq).THEOREM 3.2. Let G be an infinite simple locally finite group with a finite non-abelianp-group of automorphisms P such that(1) CG(P) is Chernikov,(2) For every α ∈ P\{1}, the set of fixed points CG(α) does not involve an infinitesimple groupThen, G is isomorphic to PSLp(k) where k is a locally finite field of characteristic q = pand P is metabelian.Proof. Since P is a finite p-group and CG(P) satisfies min-p, by [2, Lemma 2.1], Gsatisfies min-p. Then, by [4, Theorem B], G is a simple group of Lie type over a locallyfinite field k of characteristic q. Now assume that q = p. Clearly G contains a root subgroup,which is an infinite elementary abelian p-subgroup. Hence, G can not satisfy min-p. Hence,q = p, that is, G is isomorphic to a simple group of Lie type over an infinite locally finitefield of characteristic q = p.Now, by [3, Lemma 4.3], there exists a simple linear algebraic group G of adjoint type,a Frobenius map σ on G and a sequence of natural numbers ni|ni+1 such thatG =⋃i∈NOp′(Gσ ni ).By assumption, the centralizer of any non-identity element does not involve an infinitesimple group, so [2, Lemma 2.3] implies that P consists of inner-diagonal automorphismsof G. Hence, P ≤ ⋃i∈N Gσ ni . Therefore, P ≤ Gσ nj for some j ∈ N.Choose 1 = z ∈ Z(P). Clearly, P ≤ CG(z). Now, CG(z) =⋃i∈N Op′(CG(z)σ ni ).By assumption, CG(z) does not involve an infinite simple group. Now, suppose thatCG(z) involves a simple linear algebraic group H . Consider the union of fixed points ofσ ni on H , denoted by Hi = Hσ ni . Clearly, Hi ≤ Hi+1 and infinitely many of H involves finitesimple groups such that their union form an infinite locally finite simple group. Hence, weget a contradiction and we deduce CG(z) does not involve a simple linear algebraic group.By [2, Lemma 2.4], CG(z) is metabelian. Hence, P is metabelian. On the other hand, sinceP is not abelian, CG(z) is not abelian.By Proposition 3.1, G is isomorphic to PGLp(Fq). Hence, G is isomorphic to eitherPSLp(k) or PSUp(k). Following the argument in the proof of Theorem 1.2 in [2], sinceavailable at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S001708951900003XDownloaded from https://www.cambridge.org/core. Charité Universitätsmedizin Berlin, on 17 May 2021 at 11:47:30, subject to the Cambridge Core terms of use,186 KIVANÇ ERSOYthe Weyl group of PSUp(k) has no elements of order p, and PT/T embeds in the Weylgroup, PSUp(k) has no such non-abelian subgroup P. Therefore, G ∼= PSLp(k) where k isan infinite locally finite field of characteristic q = p.Then, we prove the main result of the paper:Proof of Theorem 1.3. Assume first that P is abelian. Then by Theorem 1.2, the resultfollows with |P| = p2.Now, assume P is non-abelian. By Theorem 3.2, G ∼= PSLp(k) where k is a locallyfinite field of characteristic q = p. Let 1 = z ∈ Z(P), observe that P ≤ CG(z) ≤ CG(z) whereG is the corresponding simple linear algebraic group and σ is the Frobenius map suchthat G = ⋃i∈N Op′(Gσ ni ), which exist by [3, Lemma 4.3]. Denote the maximal torus of Gcontaining z by T . By Proposition 3.1(i), CG(z)0 = T . Indeed, by [10, Corollary 1.7], T isthe unique maximal torus containing z. Since P is not abelian, CG(z)/CG(z)0 can not be 1,hence by [7, Proposition 14.20], it has exponent p. Let y be any element of CG(z)\CG(z)0.Then, Q = 〈 y, z〉 has order p2. Indeed, CG(z)0 = T , and y ∈ NG(T). Hence, y induces anelement w of order p in the Weyl group. Now, z ∈ CT (w). The computation in the proof ofTheorem 1.2 in [2] shows that indeed CT (w) has order p, hence CG(Q) = Q. This Q is therequired subgroup.ACKNOWLEDGEMENTS. This research was carried out when the author was visitingEPFL and TU Kaiserslautern. The author thanks Professor Donna Testerman and ProfessorGunter Malle for their hospitality. The author was partially supported by Mimar Sinan FineArts University Research Project Unit, with project number 2017/21. The author thanks thereferee for the suggestions and comments, which were helpful to improve the text.REFERENCES1. K. Ersoy and C. K. Gupta, Locally finite groups with centralizers of finite rank, Comm. Alg.44(12) (2016), 5074–5087.2. K. Ersoy, M. Kuzucuoǧlu and P. Shumyatsky, Locally finite groups and their subgroups withsmall centralizers, J. Algebra 481 (2017), 1–11.3. B. Hartley and M. Kuzucuoǧlu, Centralizers of elements in locally finite simple groups,Proc. London Math. Soc. 62 (1991), 301–324.4. B. Hartley and G. Shute, Monomorphisms and direct limits of finite groups of Lie type,Q. J. Math. 35(1) (1984), 49–715. O. H. Kegel and B. A. F. Wehrfritz, Strong finiteness conditions in locally finite groups,Math. Z. 117 (1970), 309–324.6. O. H. Kegel and B. A. F. Wehrfritz, Locally finite groups (North-Holland, Amsterdam,1973).7. G. Malle and D. Testerman, Linear algebraic groups and finite groups of Lie type,Cambridge Studies in Advanced Mathematics, vol. 133 (Cambridge University Press, Cambridge,2011).8. R. Steinberg, Torsion in reductive groups, Adv. Math. 15(1) (1975), 63–92.9. V. P. Shunkov, On the minimality problem for locally finite groups, Algebra Logic 9 (1970),137–151.10. T. A. Springer and R. Steinberg, “Conjugacy Classes”, in Seminar on Algebraic Groups andRelated Finite Groups, Lecture Notes in Mathematics, vol. 131 (Springer-Verlag, Berlin, 1970).available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S001708951900003XDownloaded from https://www.cambridge.org/core. 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Centralizers of p-Subgroups in Simple Locally Finite Groups

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Centralizers of p-Subgroups in Simple Locally Finite Groups

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