A result of Dixon, Evans and Smith shows that if G is a locally (soluble-by- finite) group whose proper subgroups are (finite rank) -by-abelian, then G itself has this property, i.e. the commutator subgroup of G has finite rank. It is proved here that if G is a locally (soluble-by- finite) group whose proper subgroups have MINIMAX commutator subgroup, then also the commutator subgroup G’ of G is MINIMAX. A corresponding result is proved for groups in which the commutator subgroup of every proper subgroup has finite torsion-free rank.

Process capability indices are used widely throughout the world to give a quick indication of a process capability in a format that is easy to use and understand. A process capability index CpCp that constructed for measuring the quality is an effective tool for assessing process capability, since this index can reflect whether a centering process is capable of reproducing items meeting the specifications limits. The MINIMAX approach is proposed in this paper for testing capability on the basis of precision index Cp when the producer goal is avoiding the largest possible risk. Motivations and benefits of proposing MINIMAX approach are discussed for capability test. Also, the proposed method clarified by an industrial application.

Traditionally, the MINIMAX circle location problems concern , nding a circle C in the plane such that the maximum distance from the given points to the circumference of the circle is minimized. The radius of the circle can be , xed or variable. In this paper we consider the inverse case, that is: a circle C with radius r0 is given and we want to modify the coordinate of existing points with the minimum cost such that the given circle becomes optimal. Mathematical models and some properties of the cases that circle C becomes optimal with comparing to all other circles, and circle C becomes the best circle with comparing to the circles with radius r0 are presented.