Groups with Largely Splitting Automorphisms of Orders Three and Four

Abstract

A subset X of a group G is said to be large (on the left) if, for any finite set of elements g1, . . . , gk â�� G, an intersection of the subsets giX = gix | x â�� X is not empty, that is, â�©i=1k giX â� â� . It is proved that a group in which elements of order 3 form a large subset is in fact of exponent 3. This result follows from the more general theorem on groups with a largely splitting automorphism of order 3, thus answering a question posed by Jaber amd Wagner in 1. For groups with a largely splitting automorphism I� of order 4, it is shown that if H is a normal I�-invariant soluble subgroup of derived length d then the derived subgroup H,H is nilpotent of class bounded in terms of d. The special case where I� = 1 yields the same result for groups that are largely of exponent 4. © 2003 Plenum Publishing Corporation.