The Schur multiplier and capability of pairs of some finite groups

The homological functors of a group have its origin in homotopy theory and algebraic K-theory. The Schur multiplier of a group is one of the homological functors while the Schur multiplier of pairs of groups is an extension of the Schur multiplier of a group. Besides, a pair of groups is capable if the precise center or epicenter of the pair of groups is trivial. In this research, the Schur multiplier and capability of pairs of groups of order p, p2, p3, p4, pq, p2q and p3q (where p and q are distinct odd primes) were determined. This research started with the computation of the normal subgroups of the groups. The generalized structures of the normal subgroups have been found by the assistance of Groups, Algorithms and Programming (GAP) software. By using the Sylow theorems and the results of the nonabelian tensor products, derived subgroups, centers of the groups, abelianization of the groups and the Schur multiplier of the groups, the Schur multiplier of pairs of groups of order p, p2, pq, p2q, p3, p4 and p3q were then determined. The classification of the groups had also been used in the computation of the Schur multiplier and capability of pairs of groups. The order of the epicenter of pairs of groups of order p, p2, p3, p4 and pq were also computed by using GAP software to determine the capability of pairs of the groups. The Schur multiplier of pairs of group is found to be trivial or abelian. All pairs of groups where G is isomorphic to the elementary abelian groups of order p2, p3 and p4, nonabelian group of order p3 of exponent p, direct product of two cyclic groups of order p2, semi- direct product of two cyclic groups of order p2, and direct product of cyclic group of order p and nonabelian group of order p3 of exponent p are capable. For other groups, only certain pairs of groups are capable depending on their normal subgroups

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ω is explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω

The computation of Schur multiplier and capability of pairs of groups of order p4

The homological functors of a group have its foundation in homotopy theory and algebraic K-theory. The Schur multiplier of a group is one of the homological functors while the Schur multiplier of pairs of groups is a continuation of the Schur multiplier of a group. Meanwhile, a pair of groups is capable if the precise center or epicenter of the pair of group is trivial. In this research, the Schur multiplier and capability of pairs of all abelian groups of order p4 are computed

The Schur multiplier of pairs of groups of order p(2)q

Let G, N be a pair of groups where G is a group and N is a normal subgroup of G. Then the Schur multiplier of pairs of groups G, N is a functorial abelian group M G, N. In this paper, M G, N for groups of order p2 q where p and q are prime numbers are determined

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ωis explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω