Half-factorial sets in elementary \(p\)-groups. (English) Zbl 1120.20055

Schmid, Wolfgang A.

Far East J. Math. Sci. (FJMS) 22, No. 1, 75-114 (2006).

A subset \(G_0\) of an Abelian group \(G\) is called half-factorial if all factorizations of an element in the semigroup \(B(G_0)\), consisting of zero-sum sequences of elements of \(G_0\) with juxtaposition as multiplication, are of the same length. The author studies half-factorial subsets in elementary \(p\)-groups and determines the structure of such sets with largest cardinality \(\mu(G)\). In the case when either \(p\leq 7\) or the rank \(r\) of \(G\) is \(\leq 2\), the equality \(\mu(G)=1+[r/2](p-2)+r\) is established. The author conjectures that this formula holds for all elementary \(p\)-groups.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

20K01	Finite abelian groups
11R27	Units and factorization
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations
20D60	Arithmetic and combinatorial problems involving abstract finite groups
20M14	Commutative semigroups

Keywords:

lengths of factorizations; half-factorial sets; elementary Abelian \(p\)-groups; zero-sum sequences