Factoring by hereditary periodicity forcing subsets (original) (raw)

Factorization of periodic subsets

Acta Mathematica Hungarica, 1991

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajó s's factorization theorem.

NONQUASI-PERIODIC FACTORIZATIONS FOR CERTAIN FINITE ABELIAN GROUPS

International Journal of Algebra and Computation, 2007

Hajós [2] asked if each factorization of a finite abelian group is quasi-periodic. Sands [5] exhibited a counter-example. We will show that a sizeable family of finite abelian groups admits nonquasi-periodic factorizations. We also describe a small family whose members have only quasi-periodic factorizations.

Groups admitting only quasi-periodic factorizations

We will show that if a finite abelian group is a direct product of two subgroups of relatively prime orders and one of the subgroups has prime order, then each normalized factorization of the group must be quasiperiodic.

Abelian bordered factors and periodicity

European Journal of Combinatorics, 2016

A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.