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 $g_1,\ldots,g_k$ in $G$, an intersection of the subsets $g_iX=\{g_ix\mid x\in X\}$ is not empty, that is, $\bigcap\limits_{i=1}^{k}g_iX\ne\varnothing$. 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 and Wagner in [1]. For groups with a largely splitting automorphism $\varphi$ of order 4, it is shown that if $H$ is a normal $\varphi$-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 $\varphi=1$ yields the same result for groups that are largely of exponent 4.