Kıvanç Ersoy | Free University of Berlin (original) (raw)

Papers by Kıvanç Ersoy

Journal of Algebra, Jul 1, 2017

Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of ra... more Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of rank at least 3 such that C G (A) is Chernikov and C G (a) involves no infinite simple groups for any a ∈ A #. We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of P SL p (k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that C G (A) is Chernikov and C G (A) involves no infinite simple groups for any a ∈ A # if and only if G is isomorphic to P SL p (k) for some locally finite field k of characteristic different from p and A has order p 2. 2000 Mathematics Subject Classification. 20F50, 20E36.

Communications in Algebra, Jul 6, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuolu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Communications in Algebra, Oct 10, 2012

An element of a group is called anticentral if the conjugacy class of that element is equal to th... more An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.

Journal of Group Theory, 2012

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non... more Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups. Then for any finite subgroup F consisting of K-semisimple elements in G, the centralizer C G ðF Þ has an infinite abelian subgroup A isomorphic to a direct product of Z pi for infinitely many distinct primes p i. Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups G i and F is a finite K-semisimple subgroup of G, then C G ðF Þ involves an infinite simple non-linear locally finite group provided that the finite fields k i over which the simple group G i is defined are splitting fields for L i , the inverse image of F inĜ i G i for all i A N. The groupĜ i G i is the inverse image of G i in the corresponding universal central extension group.

Glasgow Mathematical Journal, Aug 13, 2013

In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble ... more In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

arXiv: Group Theory, Aug 28, 2012

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble... more In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

A group G is called an FC-group if the conjugacy class of every element is finite. G is called a ... more A group G is called an FC-group if the conjugacy class of every element is finite. G is called a minimal non-FC-group if G is not an FC-group, but every proper subgroup of G is an FC-group. The first part of this thesis is on minimal non-FC-groups and their finitary permutational representations. Belyaev proved in 1998 that, every perfect locally finite minimal non-FC-group has non-trivial finitary permutational representation. In Chapter 3, we write the proof of Belyaev in detail. Recall that a group G is called quasi-simple if G is perfect and G/Z(G) is simple. The second part of this thesis is on finite quasi-simple groups and their coprime automorphisms. In Chapter 4, the result of Parker and Quick is written in detail: Namely; if Q is a quasi-simple group and A is a non-trivial group of coprime automorphisms of Q satisfyingM.S. - Master of Scienc

Communications in Algebra, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuolu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Glasgow Mathematical Journal, 2019

In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group ... more 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 an...

Glasgow Mathematical Journal, 2020

In Ersoy et al. [J. Algebra 481 (2017), 1-11], we have proved that if G is a locally finite group... more In Ersoy et al. [J. Algebra 481 (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 p 2 such that C G (A) is Chernikov and for every non-identity α ∈ A the centralizer C G (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. Algebra 481 (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 p 2 , the centralizer C G (A) is Chernikov and for every non-identity α ∈ A, the set of fixed points C G (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 nonabelian subgroup P of automorphisms of exponent p such that the centralizer C G (P) is Chernikov and for every non-identity α ∈ P the set of fixed points C G (α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ∼ = PSL p (k) where char k = p and P has a subgroup Q of order p 2 such that C G (P) = Q.

Communications in Algebra, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuoǧlu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Archiv der Mathematik, 2016

Journal of Group Theory, 2000

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non... more Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely letMoreover we prove that if

Glasgow Mathematical Journal, 2014

In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble ... more In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

Communications in Algebra, Oct 10, 2012

Journal of Algebra, Jul 1, 2017

Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of ra... more Let p be a prime and G a locally finite group containing an elementary abelian p-subgroup A of rank at least 3 such that C G (A) is Chernikov and C G (a) involves no infinite simple groups for any a ∈ A #. We show that G is almost locally soluble (Theorem 1.1). The key step in the proof is the following characterization of P SL p (k): An infinite simple locally finite group G admits an elementary abelian p-group of automorphisms A such that C G (A) is Chernikov and C G (A) involves no infinite simple groups for any a ∈ A # if and only if G is isomorphic to P SL p (k) for some locally finite field k of characteristic different from p and A has order p 2. 2000 Mathematics Subject Classification. 20F50, 20E36.

Communications in Algebra, Jul 6, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuolu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Communications in Algebra, Oct 10, 2012

An element of a group is called anticentral if the conjugacy class of that element is equal to th... more An element of a group is called anticentral if the conjugacy class of that element is equal to the coset of the commutator subgroup containing that element. A group is called Camina group if every element outside the commutator subgroup is anticentral. In this paper, we investigate the structure of locally finite groups with an anticentral element. Moreover, we construct some non-periodic examples of Camina groups, which are not locally solvable.

Journal of Group Theory, 2012

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non... more Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite K-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups. Then for any finite subgroup F consisting of K-semisimple elements in G, the centralizer C G ðF Þ has an infinite abelian subgroup A isomorphic to a direct product of Z pi for infinitely many distinct primes p i. Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence K ¼ fðG i ; 1Þ : i A Ng consisting of finite simple subgroups G i and F is a finite K-semisimple subgroup of G, then C G ðF Þ involves an infinite simple non-linear locally finite group provided that the finite fields k i over which the simple group G i is defined are splitting fields for L i , the inverse image of F inĜ i G i for all i A N. The groupĜ i G i is the inverse image of G i in the corresponding universal central extension group.

Glasgow Mathematical Journal, Aug 13, 2013

In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble ... more In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

arXiv: Group Theory, Aug 28, 2012

In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble... more In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

A group G is called an FC-group if the conjugacy class of every element is finite. G is called a ... more A group G is called an FC-group if the conjugacy class of every element is finite. G is called a minimal non-FC-group if G is not an FC-group, but every proper subgroup of G is an FC-group. The first part of this thesis is on minimal non-FC-groups and their finitary permutational representations. Belyaev proved in 1998 that, every perfect locally finite minimal non-FC-group has non-trivial finitary permutational representation. In Chapter 3, we write the proof of Belyaev in detail. Recall that a group G is called quasi-simple if G is perfect and G/Z(G) is simple. The second part of this thesis is on finite quasi-simple groups and their coprime automorphisms. In Chapter 4, the result of Parker and Quick is written in detail: Namely; if Q is a quasi-simple group and A is a non-trivial group of coprime automorphisms of Q satisfyingM.S. - Master of Scienc

Communications in Algebra, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuolu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Glasgow Mathematical Journal, 2019

In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group ... more 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 an...

Glasgow Mathematical Journal, 2020

In Ersoy et al. [J. Algebra 481 (2017), 1-11], we have proved that if G is a locally finite group... more In Ersoy et al. [J. Algebra 481 (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 p 2 such that C G (A) is Chernikov and for every non-identity α ∈ A the centralizer C G (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. Algebra 481 (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 p 2 , the centralizer C G (A) is Chernikov and for every non-identity α ∈ A, the set of fixed points C G (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 nonabelian subgroup P of automorphisms of exponent p such that the centralizer C G (P) is Chernikov and for every non-identity α ∈ P the set of fixed points C G (α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ∼ = PSL p (k) where char k = p and P has a subgroup Q of order p 2 such that C G (P) = Q.

Communications in Algebra, 2016

The structure of locally finite groups with an involution whose centralizer has finite rank was d... more The structure of locally finite groups with an involution whose centralizer has finite rank was described by Kuzucuoǧlu and Shumyatsky. In this paper, we investigate the structure of locally finite groups with an element of order 3 whose centralizer has finite rank. Moreover, for any p, we classify simple locally finite groups having an automorphism of order p whose set of fixed points has finite rank.

Archiv der Mathematik, 2016

Journal of Group Theory, 2000

Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non... more Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely letMoreover we prove that if

Glasgow Mathematical Journal, 2014

In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble ... more In this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

Communications in Algebra, Oct 10, 2012