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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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