Summary : A subgroup $H$ of a group $G$ is said to be permutable subgroup if and only if $HK=KH$ for every subgroup $K$ of $G$. Certainly, every normal subgroup is permutable. The converse is not true. In this research we will find all permutable subgroups of the groups of order $16$. Then, find which subgroup is permutable and not normal.

Summary : In this paper, we establish some properties of subgroup structure of finite n-ary groups.

Summary : This paper investigates the effect of employing technology in enhancing the learning outcomes of an introductory mathematics course. The students of the course were divided into two sections: the experimental section and the control section. The two sections were taught the material of the course using the traditional face-to-face method. However, the students in the experimental section had the advantage to practise on the concepts of the course using a software application. The performance of the students in both sections was compared and the results showed that the students in the experimental section outperformed their counterparts in the control section.

Summary : A subgroup $H$ of a group $G$ is said to be permutable subgroup if and only if $HK=KH$ for every subgroup $K$ of $G$. Certainly, every normal subgroup is permutable. The converse is not true. In this research we will find all permutable subgroups of the groups of order $16$. Then, find which subgroup is permutable and not normal.

Summary : The number of conjugacy classes of symmetric group, dihedral group and some nilpotent groups is obtained. Until now, it has not been obtained for all nilpotent groups. Although there are some lower bounds to this value, there is no non-trivial upper bound. This paper aims to investigate an upper bound to this number for all finite nilpotent groups. Moreover, the exact number of conjugacy classes is found for a certain case of non-abelian nilpotent groups.

Summary : Let H be a subgroup of a finite group G. Then we say that H is: bipermutable in G provided G has subgroups A and B such that G = AB, H ≤ A and H permutes with all subgroups of A and with all subgroups of B; S-bipermutable in G provided G has subgroups A and B such that G = AB, H ≤ A and H permutes with all Sylow subgroups of A and with all Sylow p-subgroups of B such that (|H|, p) = 1. In this paper we analyze the influence of bipermutable and S-bipermutable subgroups on the structure of G.

Summary : It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Summary : Based on the nonlinear control theory, the anti-synchronization between two different hyperchaotic systems is investigated. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in hyperchaotic systems. Numerical simulations are performed for to demonstrate the effectiveness of the proposed control strategy

Summary : Let A, B be subgroups of a group G and ... ≠ X ⊆ G. Then, A is said to be X-permutable with B [6] if there exists an element x ∈ X such that AB^x = B^x A. In this paper we study finite groups in which maximal subgroups of Sylow subgroups are X-permutable either with maximal subgroups or with Sylow subgroups.

Summary : A subgroup H of a group G is c-subnormal in G if G has a subnormal subgroup T such that HT=G and T 3 H ⊆ HG.[1] Using this concept, in Jaraden obtain[1] some new conditions for solubility of a finite group are given. Here we obtain local versions of these results.

Summary : It is proved that every permutable subgroup of a finite n-ary group is subnormal.

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Summary : We prove that if G is a nonsingle-element n-ary finite group that belongs to a τ-closed formation F, then G/soc(G)∈Φτ(F), where Φτ(F) is the intersection of all maximal τ-closed subformations of the τ-closed formation of n-ary groups F.

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