Abstract
A subset X of a group G is said to be large (on the left) if, for any finite set of elements g1,l... ,gkin G, an intersection of the subsets giX=gimid x in X is not empty, that is, ⋂limits{i=1} {k}giX ≠∅. 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 φ of order 4, it is shown that if His a normal φ-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 φ =1 yields the same result for groups that are largely of exponent 4.
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Makarenko, N.Y., Khukhro, E.I. Groups with Largely Splitting Automorphisms of Orders Three and Four. Algebra and Logic 42, 165–176 (2003). https://doi.org/10.1023/A:1023984509127
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DOI: https://doi.org/10.1023/A:1023984509127