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In this paper, we have studied the loops which are the semidirect products of a loop and a group. Also, the cummutant, nuclei and the center of such loops are studied.

In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur Multiplier and that of gyro-Milnor $K_{2}$ group are introduced., Comment: Accepted in Journal of Algebra and Its Applications

In this paper, the central automorphism group of the Zappa-Sz\'{e}p product of two groups is obtained.

In this paper, we have computed the automorphism groups of all groups of order $p^{2}q^{2}$, where $p$ and $q$ are distinct primes.

In this paper, we have found the automorphism group of the Zappa-Sz\'{e}p product of two groups. Also, we have computed the automorphism group of the Zappa-Sz\'{e}p product of a cyclic group of order $m$ and a cyclic group of order $p^{2}$, where $p$ is a prime., Comment: This is a first draft. Comments are welcome

In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge 5)$ of order $p^{4}$, where $p$ is a prime., Comment: Comments are welcome

In this paper, we study the properties of the associated gyrogroup ${^\circ}G$ of a given group $G$ of nilpotency class $3$. We have proved that if $3$ does not divide the order of the group $G$, then the nilpotency class of the associated gyrogroup ${^\circ}G$ is same as that of the group $G$. We have also studied the problem of abelian inner mapping group in this context., Comment: Comments are welcome

In this paper, we discuss when a class function on a finite group is a bent function. We have found a necessary condition for a class function on a finite abelian group to be bent. Also, we have found a necessary and sufficient condition for a class function on a finite cyclic group to be bent.

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $\Gamma(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article, certain properties of the commuting graph of generalized dihedral groups have been studied.

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian $p$-groups of order $p^5$. As an application we determine the capability of groups of order $p^5$., Comment: 41 pages, 5 tables

In this note, a necessary and sufficient condition for the normalizer of a core-free subgroup $H$ of a finite group $G$ to be normal in $G$ is obtained. Also, a known result of finite groups is obtained through transversal., Comment: arXiv admin note: substantial text overlap with arXiv:1402.3378, arXiv:1307.5392

In this article, we describe a key exchange protocol based on right transversals. We also describe it for general extension associated to right loops.

In this paper the notion of nilpotent right transversal and solvable right transversal has been defined. Further, it is proved that if a core-free subgroup has a generating solvable transversal or a generating nilpotent transversal, then the whole group is solvable., Comment: arXiv admin note: substantial text overlap with arXiv:1307.5392

In this note, we define a new graph $\Gamma_d(G)$ on a finite group $G$, where $d$ is a divisor of $|G|$. The vertices of $\Gamma_d(G)$ are the subgroups of $G$ of order $d$ and two subgroups $H_1$ and $H_2$ of $G$ are said to be adjacent if there exists $S_i \in \mathcal{T}(G,H_i)$ $(i=1,2)$ such that $S_1 \cong S_2$. We shall discuss the completeness of $\Gamma_d(G)$ for various groups like finite abelian groups, dihedral groups and some finite $p$-groups.

In case the stability relation is a congruence, a necessary and also a sufficient condition for its equality with the center congruence is given.

Being motivated by \cite{sk} and \cite{nms}, an example of a right loop admitting only discrete topolization is given.

Let $G$ be a finite group and $H$ be a subgroup of $G$. In this paper, we prove that if $G$ is a finite nilpotent group and $H$ a subgroup of $G$, then $H$ is normal in $G$ if and only if all normalized right transversals of $H$ in $G$ are isotopic, where the isotopism classes are formed with respect to induced right loop structures. We have also determined the number isotopy classes of transversals of a subgroup of order 2 in $D_{2p}$, the dihedral group of order $2p$, where $p$ is an odd prime.

In this paper we prove that there does not exist a subgroup $H$ of a finite group $G$ such that the number of isomorphism classes of normalized right transversals of $H$ in $G$ is four.

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of [5], which avoids the use of the classification of finite simple groups