DOI link for Non-periodic factorizations
Non-periodic factorizations book
A factorization of an abelian group is said to be bad if no factor is periodic. In Section 3.1 we show how to construct bad factorizations for certain finite cyclic groups. We also show how to extend bad factorizations of a subgroup H to bad factorizations of the group G. We shall use the terminology k-bad if there are k factors in the bad factorization. In Section 3.2 we show how representations of a finite abelian group relate
to the factorizations. In the case of finite cyclic groups, we show how the use of cyclotomic polynomials is equivalent to the use of representations. In Section 3.3 we show how representation theory can be used to show
that one factor in a factorization may be replaced by another. The idea is to replace a bad, that is, non-periodic, factor by a less bad factor that can then be studied more easily. We should like to emphasize that Theorem 3.17 (on page 58) is one of the key results of the subject.