Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The... more Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if |Cent(G)| = |Cent(G Z(G))| = n. In this paper, we characterize the 9-centralizer and the primitive 9-centralizer groups. 2010 Mathematics Subject Classification. 11A25, 20D60, 20E99. Key words and phrases. Finite groups, n-centralizer groups, primitive n-centralizer groups. *I am deeply indebted to my supervisor Prof. Ashish Kumar Das for his constant encouragement throughout my M.Phil and Ph.D career.
A finite group G is a CG-group if | Cent(G) | = | G' |+2, where G' is the commutator subg... more A finite group G is a CG-group if | Cent(G) | = | G' |+2, where G' is the commutator subgroup and Cent(G) is the set of distinct element centralizers of G. In this paper we give some results on CG-groups. We also give a negative answer to <cit.> given by K. Khoramshahi and M. Zarrin.
A finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x)... more A finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x) = C(y). On the otherhand, two elements of a group are said to be z-equivalent or in the same z-class if their centralizers are conjugate in the group. In this paper, for a finite non-abelian group, we give a necessary and sufficient condition for the number of centralizers/ z-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of z-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and z-classes of some finite groups and extend some previous results.
A group is said to be capable if it is the central factor of some group. In this paper, among oth... more A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.
In this paper, we characterize finite group G with unique proper nonabelian element centralizer. ... more In this paper, we characterize finite group G with unique proper nonabelian element centralizer. This improves [5, Theorem 1.1]. Among other results, we have proved that if C(a) is the proper non-abelian element centralizer of G for some a ∈ G, then C(a) Z(G) is the Fitting subgroup of G Z(G) , C(a) is the Fitting subgroup of G and G ′ ∈ C(a), where G ′ is the commutator subgroup of G.
A group is said to be capable if it is the central factor of some group. In this paper, among oth... more A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.
Let $G$ be a group and $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)=lbra... more Let $G$ be a group and $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)=lbrace y in G mid langle x, yrangle ; {rm is ; cyclic} rbrace$. $G$ is said to be a tidy group if $Cyc(x)$ is a subgroup for all $x in G$. We call $G$ to be a C-tidy group if $Cyc(x)$ is a cyclic subgroup for all $x in G setminus K(G)$, where $K(G)$ is the intersection of all the cyclicizers in $G$. In this note, we classify finite C-tidy groups with $K(G)=lbrace 1 rbrace$.
A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups eq... more A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. Let $p<q<r<s$ be primes. We prove that if $G$ is a Leinster group of order $p^2qr$, then $G \cong Q_{20}\times C_{19}$ or $Q_{28} \times C_{13}$. We also prove that no group of order $pqrs$ is Leinster.
A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the comm... more A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the commutator subgroup and Cent$(G)$ is the set of distinct element centralizers of $G$. In this paper we give some results on CG-groups. We also give a negative answer to \cite[Conjecture 2.3]{con} given by K. Khoramshahi and M. Zarrin.
A group G is said to be n-centralizer if its number of element centralizers | Cent(G) |= n, an F-... more A group G is said to be n-centralizer if its number of element centralizers | Cent(G) |= n, an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any n > 11, we prove that if G is an n-centralizer group, then | G Z(G) |≤ 2(n − 4) (n−4) 2 , which improves an earlier result (it is known that | G Z(G) |≤ (n − 2) for n ≤ 11). We also prove that if G is an arbitrary n-centralizer F-group, then gcd(n − 2, | G Z(G) |) 6= 1. For a finite F-group G, we show that | Cent(G) |≥ |G| 2 iff G ∼= A4, D2n (where n is odd) or an extraspecial 2-group. Among other results, for a finite group G with non-trivial center, it is proved that | Cent(G) |= |G| 2 iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.
A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)... more A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and $z$-classes of some finite groups and extend some previous results.
For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G... more For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G is called n-centralizer if | Cent(G) |= n, and primitive n-centralizer if | Cent(G) |=| Cent( G Z(G) ) |= n. In this paper, among other things, we investigate the structure of finite groups of odd order with | Cent(G) |= 9 and prove that if |G| is odd, then | Cent(G) |= 9 if and only if G Z(G) ∼= C7 o C3 or C7 × C7. Mathematics Subject Classification (2010): 20D60
Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The... more Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if |Cent(G)| = |Cent(G Z(G))| = n. In this paper, we characterize the 9-centralizer and the primitive 9-centralizer groups. 2010 Mathematics Subject Classification. 11A25, 20D60, 20E99. Key words and phrases. Finite groups, n-centralizer groups, primitive n-centralizer groups. *I am deeply indebted to my supervisor Prof. Ashish Kumar Das for his constant encouragement throughout my M.Phil and Ph.D career.
Rendiconti del Seminario Matematico della Università di Padova, 2014
Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of th... more Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of the orders of the normal subgroups of G. The group G is said to be harmonic if H(G) : jGjt(G)=s(G) is an integer. In this paper, all finite groups for which 1 H(G) 2 have been characterized. Harmonic groups of order pq and of order pqr, where p < q < r are primes, are also classified. Moreover, it has been shown that if G is harmonic and G T C 6 , then t(G) ! 6.
A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups eq... more A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. In this paper we give some new results concerning Leinster groups.
Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The... more Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if |Cent(G)| = |Cent(G Z(G))| = n. In this paper, we characterize the 9-centralizer and the primitive 9-centralizer groups. 2010 Mathematics Subject Classification. 11A25, 20D60, 20E99. Key words and phrases. Finite groups, n-centralizer groups, primitive n-centralizer groups. *I am deeply indebted to my supervisor Prof. Ashish Kumar Das for his constant encouragement throughout my M.Phil and Ph.D career.
A finite group G is a CG-group if | Cent(G) | = | G' |+2, where G' is the commutator subg... more A finite group G is a CG-group if | Cent(G) | = | G' |+2, where G' is the commutator subgroup and Cent(G) is the set of distinct element centralizers of G. In this paper we give some results on CG-groups. We also give a negative answer to <cit.> given by K. Khoramshahi and M. Zarrin.
A finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x)... more A finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x) = C(y). On the otherhand, two elements of a group are said to be z-equivalent or in the same z-class if their centralizers are conjugate in the group. In this paper, for a finite non-abelian group, we give a necessary and sufficient condition for the number of centralizers/ z-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of z-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and z-classes of some finite groups and extend some previous results.
A group is said to be capable if it is the central factor of some group. In this paper, among oth... more A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.
In this paper, we characterize finite group G with unique proper nonabelian element centralizer. ... more In this paper, we characterize finite group G with unique proper nonabelian element centralizer. This improves [5, Theorem 1.1]. Among other results, we have proved that if C(a) is the proper non-abelian element centralizer of G for some a ∈ G, then C(a) Z(G) is the Fitting subgroup of G Z(G) , C(a) is the Fitting subgroup of G and G ′ ∈ C(a), where G ′ is the commutator subgroup of G.
A group is said to be capable if it is the central factor of some group. In this paper, among oth... more A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.
Let $G$ be a group and $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)=lbra... more Let $G$ be a group and $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)=lbrace y in G mid langle x, yrangle ; {rm is ; cyclic} rbrace$. $G$ is said to be a tidy group if $Cyc(x)$ is a subgroup for all $x in G$. We call $G$ to be a C-tidy group if $Cyc(x)$ is a cyclic subgroup for all $x in G setminus K(G)$, where $K(G)$ is the intersection of all the cyclicizers in $G$. In this note, we classify finite C-tidy groups with $K(G)=lbrace 1 rbrace$.
A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups eq... more A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. Let $p<q<r<s$ be primes. We prove that if $G$ is a Leinster group of order $p^2qr$, then $G \cong Q_{20}\times C_{19}$ or $Q_{28} \times C_{13}$. We also prove that no group of order $pqrs$ is Leinster.
A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the comm... more A finite group $G$ is a CG-group if $|{\rm Cent}(G) | = | G' |+2$, where $G'$ is the commutator subgroup and Cent$(G)$ is the set of distinct element centralizers of $G$. In this paper we give some results on CG-groups. We also give a negative answer to \cite[Conjecture 2.3]{con} given by K. Khoramshahi and M. Zarrin.
A group G is said to be n-centralizer if its number of element centralizers | Cent(G) |= n, an F-... more A group G is said to be n-centralizer if its number of element centralizers | Cent(G) |= n, an F-group if every non-central element centralizer contains no other element centralizer and a CA-group if all non-central element centralizers are abelian. For any n > 11, we prove that if G is an n-centralizer group, then | G Z(G) |≤ 2(n − 4) (n−4) 2 , which improves an earlier result (it is known that | G Z(G) |≤ (n − 2) for n ≤ 11). We also prove that if G is an arbitrary n-centralizer F-group, then gcd(n − 2, | G Z(G) |) 6= 1. For a finite F-group G, we show that | Cent(G) |≥ |G| 2 iff G ∼= A4, D2n (where n is odd) or an extraspecial 2-group. Among other results, for a finite group G with non-trivial center, it is proved that | Cent(G) |= |G| 2 iff G is an extraspecial 2-group. We give a family of F-groups which are not CA-groups and extend an earlier result.
A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)... more A finite group $G$ is called an F-group if for every $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ implies that $C(x) = C(y)$. On the otherhand, two elements of a group are said to be $z$-equivalent or in the same $z$-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ $z$-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of $z$-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and $z$-classes of some finite groups and extend some previous results.
For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G... more For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G is called n-centralizer if | Cent(G) |= n, and primitive n-centralizer if | Cent(G) |=| Cent( G Z(G) ) |= n. In this paper, among other things, we investigate the structure of finite groups of odd order with | Cent(G) |= 9 and prove that if |G| is odd, then | Cent(G) |= 9 if and only if G Z(G) ∼= C7 o C3 or C7 × C7. Mathematics Subject Classification (2010): 20D60
Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The... more Given a finite group G, let Cent(G) denote the set of distinct centralizers of elements of G. The group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if |Cent(G)| = |Cent(G Z(G))| = n. In this paper, we characterize the 9-centralizer and the primitive 9-centralizer groups. 2010 Mathematics Subject Classification. 11A25, 20D60, 20E99. Key words and phrases. Finite groups, n-centralizer groups, primitive n-centralizer groups. *I am deeply indebted to my supervisor Prof. Ashish Kumar Das for his constant encouragement throughout my M.Phil and Ph.D career.
Rendiconti del Seminario Matematico della Università di Padova, 2014
Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of th... more Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of the orders of the normal subgroups of G. The group G is said to be harmonic if H(G) : jGjt(G)=s(G) is an integer. In this paper, all finite groups for which 1 H(G) 2 have been characterized. Harmonic groups of order pq and of order pqr, where p < q < r are primes, are also classified. Moreover, it has been shown that if G is harmonic and G T C 6 , then t(G) ! 6.
A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups eq... more A finite group is said to be a Leinster group if the sum of the orders of its normal subgroups equals twice the order of the group itself. In this paper we give some new results concerning Leinster groups.
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