Let cal V be a variety of groups defined by the set of laws V , it is shown that every group G possesses a uniquely determined subgroup (V^*)^*(G) of the marginal subgroup V^*(G) , which is minimal subject to being the image in G of the marginal subgroup of some cal V -marginal extension of G . (V^*)^*(G) is characteristic and is also the smallest subgroup of V^*(G) whose factor-group is cal V -capable. The Stallings sequence, and the Ganea sequence up to the left term, are also generalized to an arbitrary variety of groups. Finally if { cal N}_c is the variety of nilpotent groups of class at most c (c geq 1) , and G is an { cal N}_c -capable group and also G /gamma_{c+1}(G) is of finite exponent, then it is shown that the exponent of the i^{th} -centre , Z_i(G) , for all 1 leq i leq c , divides the exponent of G gamma_{c+1}(G) .

Keywords