Grushin problems and control theory: Formulation and examples

In this paper we give a new formulation of an abstract control problem in terms of a Grushin problem, so that we will reformulate all notions of controllability, observability and stability in a new form that gives readers an easy interpretation of these notions

The schur multipliers, nonbelian tensor squares and capability of some finite p-groups

The homological functors and nonabelian tensor product have its roots in algebraic K-theory as well as in homotopy theory. Two of the homological functors are the Schur multiplier and nonabelian tensor square, where the nonabelian tensor square is a special case of the nonabelian tensor product. A group is said to be capable if it is a central factor group. In this research, the Schur multiplier, nonabelian tensor square and capability for some groups of order p3, p4, p5 and p6 are determined. An algebraic computation of the center, derived subgroups, abelianization, Schur multipliers, nonabelian tensor squares and capability of the groups are determined with the assistance of Groups, Algorithms and Programming (GAP) software. Using the results of the center, derived subgroups and abelianization, the Schur multiplier, nonabelian tensor square and capability for the groups are determined. The nonabelian tensor squares and capability are also determined using the results of the Schur multipliers. The Schur multiplier of each of the groups considered is found to be trivial or abelian. The results show that the nonabelian tensor square of the groups are always abelian. In addition, a group has been shown to be capable if it has a nontrivial kernel or it is an extra-special p-group with exponent p

Series of a construction related to the non-abelian tensor square of groups

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove that the derived subgroup $\nu(G)'$ is a central product of three normal subgroups of $\nu(G)$, all isomorphic to the non-abelian tensor square $G \otimes G$. As a consequence, we describe the structure of each term of the derived and lower central series of the group $\nu(G)$

The nonabelian tensor square of a Bieberbach group with symmetric point group of order six

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is given on the Bieberbach groups with symmetric point group of order six. The nonabelian tensor square of a group is a well known homological functor which can reveal the properties of a group. With the method developed for polycyclic groups, the nonabelian tensor square of one of the Bieberbach groups of dimension four with symmetric point group of order six is computed. The nonabelian tensor square of this group is found to be not abelian and its presentation is constructed