Nor Muhainiah Mohd Ali

Relative commutativity degree of some dihedral groups

The commutativity degree of a finite group G was introduced by Erdos and Turan for symmetric groups, finite groups and finite rings in 1968. The commutativity degree, P(G), is defined as the probability that a random pair of elements in a group commute. The relative commutativity degree of a group G is defined as the probability for an element of subgroup, H and an element of G to commute with one another and denoted by P(H,G). In this research the relative commutativity degree of some dihedral groups are determined.

Compatible pair of nontrivial actions for some cyclic groups of 2-power order

Compatible actions play a very important verification before the nonabelian tensor product can be computed. This paper gives the some exact number of compatible pairs of actions for some cyclic groups of 2-power order. Some necessary and sufficient numbers of theoretical conditions for a pair of cyclic groups of 2-power order with nontrivial actions which act compatibly on each other are used to investigate some properties in order to find the exact number of compatible pairs of actions. Algorithms in Groups, Algorithms and Programming (GAP) software are used to create more examples on selected cases. New results on compatible pair of nontrivial actions of order two and four for cyclic groups of 2-power order are presented in this paper.

The productivity degree of two subgroups of dihedral groups

The commutativity degree of a group is the probability that two randomly chosen elements of G commute. The concept of commutativity degree is then extended to the relative commutativity degree of a group, which is defined as the probability for an element of H and an element of G to commute to one another. In this research, the relative commutativity degree concept is extended to the probability of a product of two subgroups of a group G productivity degree of two subgroups of a group G which is defined as the ratio of the order of the intersection of HK and KH with the order of their union where H and K are the subgroups of a group G. This research focuses only on the dihedral groups.

The generalization of the Schur multipliers of Bieberbach groups

The Schur multiplier is the second homology group of a group. It has been found to be isomorphic to the kernel of a homomorphism which maps the elements in the exterior square of the group to the elements in its derived subgroup. Meanwhile, a Bieberbach group is a space group which is a discrete cocompact group of isometries of oriented Euclidean space. In this research, the Schur multipliers of Bieberbach groups with cyclic point group of order two of finite dimension are computed.

ON THE CAPABILITY OF FINITELY GENERATED NON-TORSION GROUPS OF NILPOTENCY CLASS 2

A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

The relative commutativity degree of noncyclic subgroups of nonabelian metabelian groups of order at most 14

A metabelian group is a group G that have at least an abelian normal subgroup N such that the quotient group GN is also abelian. The concept of commutativity degree plays as important role in determining the abelianness of the group. This concept has been extended to the relative commutativity degree of a subgroup H of a group G which defined as the probability that an element of H commutes with an element of G. In this paper, the relative commutativity degree for noncyclic subgroups of nonabelian metabelian groups of order at most 14 are determined.

The multiplicative degree of cyclic subgroups of nonabelian metabelian groups of order less than 24

A group G is metabelian if and only if there is an abelian normal subgroup N such that the quotient group, GN is abelian. The probability that a random element commutes with another random element in G is called the commutativity degree of a group. Furthermore, the relative commutativity degree of a subgroup H is defined as the probability that a random element of subgroup, H commutes with another random element of a group G. The concept of relative commutativity degree has been extended to the multiplicative degree of a group G, which is defined as the probability that the product of two randomly selected elements from a group G, is in H. In this paper, the multiplicative degree of cyclic subgroups of nonabelian metabelian groups of order less than 24 are determined.

Multiplicative degree of some dihedral groups

Let G be a group and H any subgroup of G. The commutativity degree of a finite group G is defined as the probability that a pair of elements x and y, chosen randomly from a group G, commute. The concept of commutativity degree has been extended to the relative commutativity degree of a subgroup H, which is defined as the probability that a random element of a subgroup, H commutes with another random element of a group G. This research extends the concept of relative commutativity degree to the multiplicative degree of a group G, which is defined as the probability that the product of a pair of elements x, y chosen randomly from a group G, is in H. This research focuses on some dihedral groups.

The cubed commutativity degree of dihedral groups

Let G be a finite group. The commutativity degree of a group is the probability that a random pair of elements in the group commute. Furthermore, the n-th power commutativity degree of a group is a generalization of the commutativity degree of a group which is defined as the probability that the n-th power of a random pair of elements in the group commute. In this research, the n-th power commutativity degree for some dihedral groups is computed for the case n equal to 3, called the cubed commutativity degree.

The Schur multiplier of pairs of groups of order p3q

Let (G, N) be a pair of groups in which N is a normal subgroup of G. Then, the Schur multiplier of pairs of groups (G, N), denoted by M (G, N), is an extension of the Schur multiplier of a group G, which is a functorial abelian group. In this research, the Schur multiplier of pairs of all groups of order p3q where p is an odd prime and p < q is determined.