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Small squaring and cubing properties for finite groups. (English) Zbl 0728.20020
Let G be a finite group. If K is a subset of G with k elements, K is a k- set. If $$m\geq 2$$ is a natural number, $$K^ m:=\{a_ 1a_ 2...a_ m|$$ $$a_ i\in K$$, $$1\leq i\leq m\}$$. This paper follows a series of articles by the first two authors in considering the problem of determining the structure of G if certain constraints on the numbers $$| K^ m|$$ are imposed. G is said to have the small squaring property on k-sets if $$| K^ 2| <k^ 2$$ for all k-element subsets K of G; the small cubing property on k-sets is defined in a similar way.
A complete characterization of the finite groups G having the small squaring property for 3-sets is obtained if G is not a non-Abelian 2- group of exponent 4. It is also shown that even stronger restrictions on $$| K^ 2|$$ have a strong impact on the structure of G: the only non-Abelian group G having the property that $$| K^ 2| <7$$ for all 3-sets of G is $$S_ 3$$. The small cubing property is also considered on 2-sets: a group G of odd order satisfies this property iff G is Abelian or is non-Abelian of exponent 3. A characterization of those finite groups G satisfying $$| K^ 3| <6$$ for all 2-sets K of G is also obtained.
The proofs use - as one can expect - a nice combination of general results with ad-hoc arguments involving the structure of elements of the sets $$K^ m$$. At least from the point of view of the length of the arguments involved, it seems that the difficulties increase with $$| K|$$ more rapidly than with m, at least for small values of $$| K|$$ and m.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20E34 General structure theorems for groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 11P99 Additive number theory; partitions
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