ismail guloglu - Academia.edu (original) (raw)
Papers by ismail guloglu
International Journal of Algebra and Computation, 2020
Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Form... more Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see te...
Communications in Algebra, 2015
Communications in Algebra, 2014
We prove that a finite solvable group G admitting a Frobenius group F H of automorphisms of copri... more We prove that a finite solvable group G admitting a Frobenius group F H of automorphisms of coprime order with kernel F and complement H so that [G, F ] = G and C C G (F) (h) = 1 for every 1 = h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.
Hacettepe Journal of Mathematics and Statistics, 2014
Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finit... more Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph-which we name as a truncated tree associated with A-whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with κ(A) = n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.
TURKISH JOURNAL OF MATHEMATICS, 2014
A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup ... more A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that F H/[F, F ] is a Frobenius group with Frobenius kernel F/[F, F ]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms F H aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.
Journal of Algebra, 2014
Abstract A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal s... more Abstract A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that F H / [ F , F ] is a Frobenius group with Frobenius kernel F / [ F , F ] . Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide | F H | . It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C V ( H ) of the fixed-point subspace of H by g ( m ) = 3 + [ log 2 ( m + 1 ) ] . It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g ( r ) , where r is the sectional rank of C G ( H ) . As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to | G | , a bound for the p-length of G is obtained in terms of the rank of a Hall p ′ -subgroup of C G ( φ ) . Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic).
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2013
In this study we have obtained some sufficient conditions for the Taketa inequality namely dlðGÞ ... more In this study we have obtained some sufficient conditions for the Taketa inequality namely dlðGÞ jcdðGÞj for finite solvable groups G.
Proceedings of the Edinburgh Mathematical Society, 2011
Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding ... more Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.
Journal of Pure and Applied Algebra, 1996
Kumar et al. (1985) introduced the concept of generalized bent functionsj': z: + Z, where q is a ... more Kumar et al. (1985) introduced the concept of generalized bent functionsj': z: + Z, where q is a positive integer > 1, and gave constructions for such functions for every possible value of L/ and II other than II odd and y = 2 (mod 4). Furthermore, they have shown the non-existence in the remaining case under certain sufficient conditions. The main purpose of this paper is to understand the extent of the set of parameters for which no generalized bent functions exist. In particular. the non-existence of Bent functions on L:,,. with p = 7 (mod 8) and r 2 1 is examined. The result obtained generalizes recent works of Bi (1991) and Pei (1993).
![Research paper thumbnail of Erratum to Action of a Frobenius-like group with fixed-point free kernel [J. Group Theory (2014), DOI 10.1515/jgt-2014-0002]](https://attachments.academia-assets.com/97264310/thumbnails/1.jpg)
](Journal of Group Theory, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessi... more We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that OEF; h D F for all nonidentity elements h 2 H. We prove that any irreducible nontrivial FH-module for a Frobeniuslike group FH of odd order over an algebraically-closed field has an H-regular direct summand if either F is fixed-point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.
Journal of Group Theory, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup
Journal of Algebra and Its Applications, 2013
In this paper we study the structure of a finite group G admitting a solvable group A of automorp... more In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ CG(A) of prime order or of order 4, every conjugate of x in G is also contained in CG(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.
Journal of Algebra, 1979
The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group ... more The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group containing a 3-central element r of order 3 such that C,(rr)j(w> is isomorphic to A,. If G is not 3-~o~~a~ then G is isomorphic to He. This yields the following COKQLLARY. Let G be a jkite, non-abelian, sim@e group containing alz element T of order 3 such that Co(~))/(~) is isomorphic to A, or S,. Assume f~~~the~mQ~e at there exists in G an elementary abelian subgroup of order 9 all rzon-identity elements of which me conjugate to rr. Then G is isomorphic to He. The proof of the Theorem will be based on the following YPOTI-IESIS. Let G be a simple group containing a standard subgroup A stich A/Z(A) is isomorphic to PSL(3, 4). Then G : ZF i.~omoqphic to He or SW. OY Ol$r~ Here He, Suz, ON denote the sporadic simple groups of orders 1P33527~17, 21337527.1H "13, 2g345721 1.19.31 respectively discovered by O'Nan. In [3] Gheng proves the Hypothesis under one of the following further conditions: (i) The Sylow 2-subgroup of Z(A) is non-trivial. (ii) 211 does not divide the order of G. The full result might be established by now. The notation is standard (see [q and [lo]). In particular A, and S, denote respectively the alternating and symmetric group of degree n. PSL(n, q) is the projective special linear group of dimension n on a field with p elements. Z, denotes the cyclic group of order n and E,n the elementary abelian p-group of 261
Journal of Algebra, 2008
Gülin Ercan andİsmail Ş. Güloglu Theorem B. Let H be a group of odd order not divisible by 3. Sup... more Gülin Ercan andİsmail Ş. Güloglu Theorem B. Let H be a group of odd order not divisible by 3. Suppose that its Carter subgroups have a normal complement G. If C is a Carter subgroup of H, then f (G) ≤ 2(2 (C) − 1). Theorem C. Let G be a group of odd order not divisible by 3. If C is a Carter subgroup of G, then f (G) ≤ 4(2 (C) − 1) − (C). Except for the following, the notation and terminology are as in [2]. For any group G,G denotes the Frattini factor group of G. Let K be a group acting on finite solvable groups H and G. We say (K on G) and (K on H) are weakly equivalent if each nontrivial irreducible section of (K on G) is K-isomorphic to an irreducible section of (K on H) and vice versa. We write (K on H) ≡ w (K on G) if (K on H) is weakly equivalent to (K on G). Some Remarks Let K, L, G and H be groups. (a) If (K on G) ≡ w (K on H), then (L on G) ≡ w (L on H) for each L ≤ K. (b) Let L act on K and K act on G and H. If (K on G) ≡ w (K on H), then (K on G) ≡ w (K on H) for each ∈ L. (c) Let V be a completely reducible kG-module for a field k and let L act on G. Let ∈ L and V denote the kG-module with respect to (G on V). Assume that (G on V) ≡ w (G on V). Let M ≤ G such that M is < >-invariant, and W be the sum of all irreducible kG-submodules of V on which M acts nontrivially. Then W = W # = W as subspaces where W # stands for the sum of all irreducible kG-submodules of V on which M acts nontrivially. Note that W and W need not be isomorphic as kG-modules. Lemma 1: Let S < α > be a group where S S < α >, S is an s-group for some prime s, Φ(S) ≤ Z(S), < α > is cyclic of order p for an odd prime p. Suppose that V is a kS < α >-module for a field k of characteristic different from s. Then C V (α) = 0 if one of the following is satisfied: (i) [Z(S), α] is nontrivial on V .
Journal of Algebra, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessi... more We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F, h] = F for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial F Hmodule for a Frobenius-like group F H of odd order over an algebraically closed field has an H-regular direct summand if either F is fixed point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.
International Journal of Algebra and Computation, 2012
Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose tha... more Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p′-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.
Archiv der Mathematik, 1991
Archiv der Mathematik, 1990
International Journal of Algebra and Computation, 2020
Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Form... more Let [Formula: see text] be a finite solvable group and [Formula: see text] be a subgroup of [Formula: see text]. Suppose that there exists an [Formula: see text]-invariant Carter subgroup [Formula: see text] of [Formula: see text] such that the semidirect product [Formula: see text] is a Frobenius group with kernel [Formula: see text] and complement [Formula: see text]. We prove that the terms of the Fitting series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the Fitting series of [Formula: see text], and the Fitting height of [Formula: see text] may exceed the Fitting height of [Formula: see text] by at most one. As a corollary it is shown that for any set of primes [Formula: see text], the terms of the [Formula: see text]-series of [Formula: see text] are obtained as the intersection of [Formula: see text] with the corresponding terms of the [Formula: see text]-series of [Formula: see text], and the [Formula: see te...
Communications in Algebra, 2015
Communications in Algebra, 2014
We prove that a finite solvable group G admitting a Frobenius group F H of automorphisms of copri... more We prove that a finite solvable group G admitting a Frobenius group F H of automorphisms of coprime order with kernel F and complement H so that [G, F ] = G and C C G (F) (h) = 1 for every 1 = h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.
Hacettepe Journal of Mathematics and Statistics, 2014
Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finit... more Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. In this paper we will consider the special case where all division rings are exactly the field K. All such finite dimensional semisimple algebras arise as a finite dimensional Leavitt path algebra. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely detemined specific graph-which we name as a truncated tree associated with A-whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with κ(A) = n. Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and determine the number of distinct Leavitt path algebras of a line graph with a given number of vertices.
TURKISH JOURNAL OF MATHEMATICS, 2014
A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup ... more A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that F H/[F, F ] is a Frobenius group with Frobenius kernel F/[F, F ]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms F H aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.
Journal of Algebra, 2014
Abstract A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal s... more Abstract A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F called kernel which has a nontrivial complement H such that F H / [ F , F ] is a Frobenius group with Frobenius kernel F / [ F , F ] . Suppose that a Frobenius-like group FH acts faithfully by linear transformations on a vector space V over a field of characteristic that does not divide | F H | . It is proved that the derived length of the kernel F is bounded solely in terms of the dimension m = dim C V ( H ) of the fixed-point subspace of H by g ( m ) = 3 + [ log 2 ( m + 1 ) ] . It follows that if a Frobenius-like group FH acts faithfully by coprime automorphisms on a finite group G, then the derived length of the kernel F is at most g ( r ) , where r is the sectional rank of C G ( H ) . As an application, for a finite solvable group G admitting an automorphism φ of prime order coprime to | G | , a bound for the p-length of G is obtained in terms of the rank of a Hall p ′ -subgroup of C G ( φ ) . Earlier results of this kind were known only in the special case when the complement of the acting Frobenius-like group was assumed to have prime order and its fixed-point subspace (or subgroup) was assumed to be one-dimensional (or have all Sylow subgroups cyclic).
Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2013
In this study we have obtained some sufficient conditions for the Taketa inequality namely dlðGÞ ... more In this study we have obtained some sufficient conditions for the Taketa inequality namely dlðGÞ jcdðGÞj for finite solvable groups G.
Proceedings of the Edinburgh Mathematical Society, 2011
Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding ... more Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.
Journal of Pure and Applied Algebra, 1996
Kumar et al. (1985) introduced the concept of generalized bent functionsj': z: + Z, where q is a ... more Kumar et al. (1985) introduced the concept of generalized bent functionsj': z: + Z, where q is a positive integer > 1, and gave constructions for such functions for every possible value of L/ and II other than II odd and y = 2 (mod 4). Furthermore, they have shown the non-existence in the remaining case under certain sufficient conditions. The main purpose of this paper is to understand the extent of the set of parameters for which no generalized bent functions exist. In particular. the non-existence of Bent functions on L:,,. with p = 7 (mod 8) and r 2 1 is examined. The result obtained generalizes recent works of Bi (1991) and Pei (1993).
Journal of Group Theory, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessi... more We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that OEF; h D F for all nonidentity elements h 2 H. We prove that any irreducible nontrivial FH-module for a Frobeniuslike group FH of odd order over an algebraically-closed field has an H-regular direct summand if either F is fixed-point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.
Journal of Group Theory, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup
Journal of Algebra and Its Applications, 2013
In this paper we study the structure of a finite group G admitting a solvable group A of automorp... more In this paper we study the structure of a finite group G admitting a solvable group A of automorphisms of coprime order so that for any x ∈ CG(A) of prime order or of order 4, every conjugate of x in G is also contained in CG(A). Under this hypothesis it is proven that the subgroup [G, A] is solvable. Also an upper bound for the nilpotent height of [G, A] in terms of the number of primes dividing the order of A is obtained in the case where A is abelian.
Journal of Algebra, 1979
The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group ... more The purpose of this paper is to prove the THEOREM. Let G be a jinite, noaz-abelian, simple group containing a 3-central element r of order 3 such that C,(rr)j(w> is isomorphic to A,. If G is not 3-~o~~a~ then G is isomorphic to He. This yields the following COKQLLARY. Let G be a jkite, non-abelian, sim@e group containing alz element T of order 3 such that Co(~))/(~) is isomorphic to A, or S,. Assume f~~~the~mQ~e at there exists in G an elementary abelian subgroup of order 9 all rzon-identity elements of which me conjugate to rr. Then G is isomorphic to He. The proof of the Theorem will be based on the following YPOTI-IESIS. Let G be a simple group containing a standard subgroup A stich A/Z(A) is isomorphic to PSL(3, 4). Then G : ZF i.~omoqphic to He or SW. OY Ol$r~ Here He, Suz, ON denote the sporadic simple groups of orders 1P33527~17, 21337527.1H "13, 2g345721 1.19.31 respectively discovered by O'Nan. In [3] Gheng proves the Hypothesis under one of the following further conditions: (i) The Sylow 2-subgroup of Z(A) is non-trivial. (ii) 211 does not divide the order of G. The full result might be established by now. The notation is standard (see [q and [lo]). In particular A, and S, denote respectively the alternating and symmetric group of degree n. PSL(n, q) is the projective special linear group of dimension n on a field with p elements. Z, denotes the cyclic group of order n and E,n the elementary abelian p-group of 261
Journal of Algebra, 2008
Gülin Ercan andİsmail Ş. Güloglu Theorem B. Let H be a group of odd order not divisible by 3. Sup... more Gülin Ercan andİsmail Ş. Güloglu Theorem B. Let H be a group of odd order not divisible by 3. Suppose that its Carter subgroups have a normal complement G. If C is a Carter subgroup of H, then f (G) ≤ 2(2 (C) − 1). Theorem C. Let G be a group of odd order not divisible by 3. If C is a Carter subgroup of G, then f (G) ≤ 4(2 (C) − 1) − (C). Except for the following, the notation and terminology are as in [2]. For any group G,G denotes the Frattini factor group of G. Let K be a group acting on finite solvable groups H and G. We say (K on G) and (K on H) are weakly equivalent if each nontrivial irreducible section of (K on G) is K-isomorphic to an irreducible section of (K on H) and vice versa. We write (K on H) ≡ w (K on G) if (K on H) is weakly equivalent to (K on G). Some Remarks Let K, L, G and H be groups. (a) If (K on G) ≡ w (K on H), then (L on G) ≡ w (L on H) for each L ≤ K. (b) Let L act on K and K act on G and H. If (K on G) ≡ w (K on H), then (K on G) ≡ w (K on H) for each ∈ L. (c) Let V be a completely reducible kG-module for a field k and let L act on G. Let ∈ L and V denote the kG-module with respect to (G on V). Assume that (G on V) ≡ w (G on V). Let M ≤ G such that M is < >-invariant, and W be the sum of all irreducible kG-submodules of V on which M acts nontrivially. Then W = W # = W as subspaces where W # stands for the sum of all irreducible kG-submodules of V on which M acts nontrivially. Note that W and W need not be isomorphic as kG-modules. Lemma 1: Let S < α > be a group where S S < α >, S is an s-group for some prime s, Φ(S) ≤ Z(S), < α > is cyclic of order p for an odd prime p. Suppose that V is a kS < α >-module for a field k of characteristic different from s. Then C V (α) = 0 if one of the following is satisfied: (i) [Z(S), α] is nontrivial on V .
Journal of Algebra, 2014
We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessi... more We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F, h] = F for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial F Hmodule for a Frobenius-like group F H of odd order over an algebraically closed field has an H-regular direct summand if either F is fixed point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.
International Journal of Algebra and Computation, 2012
Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose tha... more Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p′-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.
Archiv der Mathematik, 1991
Archiv der Mathematik, 1990