On some Abelian p-groups and their capability

A group is said to be capable if it is a central factor group; equivalently, if and only if a group is isomorphic to the inner automorphism group of another group. In this research, the capability of some abelian pgroups which are groups of order p4 and p5, where p is an odd prime are determined. The capability of the groups is determined by using the classifications of the groups

On the computations of some homological functors of 2-Engel groups of order at most 16

The homological functors including J (G) , ∇ (G) , exterior square, the Schur multiplier, Δ (G) , the symmetric square and J (G) of a group were originated in homotopy theory. The nonabelian tensor square which is a special case of the nonabelian tensor product is vital in the computations of the homological functors of a group. It was introduced by Brown and Loday in 1987. The nonabelian tensor square G⊗G of a group G is generated by the symbols g ⊗ h, for all g,h∈G subject to the relations gg′⊗h=(gg′⊗gh)(g⊗h) and g⊗hh′=(g⊗h)(hg⊗hh′), for all g,g′,h,h′ ∈G where g g′ = gg′g−1 . In this paper, the computations of nonabelian tensor squares and some homological functors of all 2-Engel groups of order at most 16 are done. Groups, Algorithms and Programming (GAP) software has been used to assist and verify the results

The Co-Prime Probability of p-Groups / Norarida Abdul Rhani ...[et al.]

The commutativity degree of Gcanbe used to measure how close a group is to be abelian. This concept has been extended to the probability that two random integers are co-prime. Two integers s and tare said to be co-prime if their greatest common divisor is equal to one. This concept hasbeen further extended to the co-prime probability of G, where the probability of the order of a random pair of elements in the group are co-prime.In this paper, the co-prime probability for all p-groups, where pis prime number isdetermine

Graph coloring using commuting order product prime graph

The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph’s general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs

The computation of some properties of additive and multiplicative groups of integers modulo n using C++ programming

This research is focused on two types of finite abelian groups which are the group of integers under addition modulo , and the group of integers under multiplication modulo , where is any positive integer at most 200. The computations of some properties of the group including the order of the group, the order and inverse of each element, the cyclic subgroups, the generators of the group, and the lattice diagrams get more complicated and time consuming as n increases. Therefore, a special program is needed in the computation of these properties. Thus in this research, a program has been developed by using Microsoft Visual C++ Programming. This program enables the user to enter any positive integer at most 200 to generate answers for the properties of the groups

Capability and homological functors of infinite two - generator groups of nilpotency class two

A group is called capable if it is a central factor group. Baer characterized finitely generated abelian groups which are capable as those groups which have two or more factors of maximal order in their direct decomposition. The capability of groups have been determined for infinite metacyclic groups and for 2-generator p-group of nilpotency class two (p prime). The remaining case for capability of 2-generator group of nilpotency class two is the infinite case where the groups have been classified by Sarmin in 2002. Let R be the class of infinite 2-generator groups of nilpotency class two. Using their classification and non-abelian tensor squares, the capability of groups in R are determined. Brown and Loday in 1984 and 1987 introduced the nonabelian tensor square of a group to be a special case of the nonabelian tensor product which has its origin in algebraic K-theory as well as in homotopy theory. The homological functors have been determined for infinite metacyclic groups and non-abelian 2-generator p-groups of nilpotency class two. Therefore, the homological functors including the exterior square, the symmetric square and the Schur multiplier of groups in R will also be determined in this research

On Some Problems in Group Theory of Probabilistic Nature

The determination of the abelianness of a nonabelian group has been introduced for symmetric groups by Erdos and Turan in 1968. In 1973, Gustafson did the same for finite groups while MacHale determined the abelianness for finite rings in 1974. Basic probability theory will be used in connection with group theory. This paper will focus on the 2-generator 2-groups of nilpotency class 2 based on the classification that has been done by Kappe et.al in 1999. In this paper some results on Pn(G), the probability that the nth power of a random element in a group G commutes with another random element from the same group, will be presented

Abelianization of some finite metacyclic 2-groups

A group G is metacyclic if it contains a cyclic normal subgroup K such that G/K is also cyclic. Finite metacyclic groups can be presented with two generators and three defining relations. In this work, we determine the structures of the derived subgroup, abelianization and itsWhitehead's quadratic functor of some finite nonabelian metacyclic 2-groups based on their classification done by Beuerle in 2005. In addition, the nonabelin tensor product for some cyclicgroups is determined

Formal language theory and DNA

Formal language theory is a branch of applied group theory that is denoted to the study of finite strings called language over some symbols chosen from a prescribed finite set called alphabet. A new manner of relating formal language theory to the study of informational macromolecules is initiated. A language is associated with each pair of sets where the first set consists of double-stranded DNA molecules and the second set consists of the recombinational behaviors allowed by specified classes of enzymatic activities. The scope of this research is on the potential effect of sets of restriction enzymes and ligase that allow DNA molecules to be cleaved and reassociated to produce further molecules. The associated languages are analysed by means of a new generative formalism called a splicing system. Splicing systems were originally developed as a mathematical or dry model of the generative of DNA molecules in the presence of appropriate restriction enzymes and a ligase. A significant subclass of these languages, which we call the persistent splicing languages, is shown to coincide with a class of regular languages which have been previously study in other contexts: the strictly locally testable languages. The relationship between the family SH of simple splicing language and the family of strictly locally testable languages is clarified. This study initiates the formal analysis of the generative power of recombinational behaviors in general. The splicing system formalism allows observations to be made concerning the generative power of general recombination and also of sets of enzymes that include general recombination

On the computation of some homological functors of 2-engel groups of order at most 16

The homological functors including J (G) , ∇(G) , exterior square, the Schur multiplier, Δ(G) , the symmetric square and