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Factorization of periodic subsets. II. (English) Zbl 0726.20036

Let G be a finite abelian group written multiplicatively with identity e and let B, \(A_ 1,...,A_ n\subseteq G\). If each \(b\in B\) can be expressed uniquely in the form \(b=a_ 1...a_ n\), \(a_ 1\in A_ 1,...,a_ n\in A_ n\) and if each product \(a_ 1...a_ n\in B\) then \(A_ 1...A_ n\) is called a factorization of the subset B. The factorization is called normed if \(e\in A_ 1\cap...\cap A_ n\). The subset B is said to be periodic if there is a \(g\in G\) with \(gB=B\) and \(g\neq e\). The paper contains two results, both concluding that one of the factors is a subgroup. The first result is the following Theorem 1. Let A be a periodic subset of the finite abelian p-group G and let \(A=A_ 1...A_ n\) be a normed factorization. If for each i, \(1\leq i\leq n| A_ i|\) is prime or there is a subgroup \(H_ i\) of G with \(| A_ i| =| H_ i| \geq 3\) and \(| A_ i\cap H_ i| \geq | A_ i| -1\), then one of the factors \(A_ i\) is a subgroup.
Reviewer: R.F.Tichy (Graz)

MSC:

20K01 Finite abelian groups
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
11H31 Lattice packing and covering (number-theoretic aspects)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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