on varieties and the direct limit of groups

Leedham-green and mckay [ acta math.137( 1967) 99-150] introduced the generalized version of the baer-invariant of a group with respect to two varieties of groups. a group g is called capable if there exists a group h such that g=h/z(h). in this paper , we generalize some properties of capability of direct product of groups with respect to two varieties of groups and direct limits.moreover, we survey some properties of the baer - invariant of a pair of groups with respect to two varieties of groups and direct limits.