arXiv subjects
Publications and source records attributed to Sekhar Jyoti Baishya.
We have classified, upto isoclinism, certain groups with a given central factor. As an application, we classify, upto isoclinism, groups having at the most nine element centralizers. Among other results of independent interest, we have classified, upto isoclinism, groups having a central factor of order $p^3$, $p$ a prime. All these improves some previous results.
In this paper, among other results, we give some sufficient conditions for every non-abelian subgroup of a group to be isoclinic with the group itself. It is also seen that under certain conditions, two groups have same number of element centralizers implies they are isoclinic. We prove that if $G$ is any group having $4, 5, 7$ or $9$ element centralizers and $H$ is any non-abelian subgroup of $G$, then $\mid \Cent(G)\mid=\mid \Cent(H)\mid$ and $ G' \cong H' \cong C_2, C_3, C_5$ or $C_7$ respectively. Furthermore, it is proved that if $G$ is any group having $n \in \lbrace 4, 5, 6, 7, 9 \rbrace$ element centralizers, then $\mid G' \mid=n-2$.
A group $G$ is said to be $n$-centralizer if its number of element centralizers $\mid \Cent(G)\mid=n$, 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 non-abelian $n$-centralizer group $G$, we prove that $\mid \frac{G}{Z(G)}\mid \leq (n-2)^2$, if $n \leq 12$ and $\mid \frac{G}{Z(G)}\mid \leq 2(n-4)^{{log}_2^{(n-4)}}$ otherwise, which improves an earlier result. We prove that if $G$ is an arbitrary non-abelian $n$-centralizer F-group, then gcd$(n-2, \mid \frac{G}{Z(G)}\mid) \neq 1$. For a finite F-group $G$, we show that $\mid \Cent(G)\mid \geq \frac{\mid G \mid}{2}$ iff $G \cong A_4 $, an extraspecial $2$-group or a Frobenius group with abelian kernel and complement of order $2$. Among other results, for a finite group $G$ with non-trivial center, it is proved that $\mid \Cent(G)\mid = \frac{\mid G \mid }{2}$ 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 $x, y \in G \setminus Z(G)$, $C(x) \leq C(y)$ 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.
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.
In this paper, we characterize finite group $G$ with unique proper non-abelian element centralizer. This improves \cite[Theorem 1.1]{nab}. Among other results, we have proved that if $C(a)$ is the proper non-abelian element centralizer of $G$ for some $a \in G$, then $\frac{C(a)}{Z(G)}$ is the Fitting subgroup of $\frac{G}{Z(G)}$, $C(a)$ is the Fitting subgroup of $G$ and $G' \in C(a)$, 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 other results we have characterized capable groups of order $p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, {\em On the non abelian tensor square and capability of groups of order $p^2q$}, Arch. Math., \textbf{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 2.10 of S. M. Jafarian Amiri, H. Madadi and H. Rostami, {\em On $F$-groups with the central factor of order $p^4$}, Math. Slovaca, \textbf{67} (5) (2017), 1147--1154.
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.
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 $|Cent(G)|=|Cent(\frac{G}{Z(G)})|=n$. In this paper, we characterize the $9$-centralizer and the primitive $9$-centralizer groups.