Groups with Largely Splitting Automorphisms of Orders Three and Four

A subset X of a group G is said to be large (on the left) if, for any finite set of elements g₁,l... ,g_kin G, an intersection of the subsets g_iX=g_imid x in X is not empty, that is, limits_{i=1}^{k}g_iX . 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 of order 4, it is shown that if His a normal -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 =1 yields the same result for groups that are largely of exponent 4.