Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Hajós’ Theorem is also called the »Second main theorem for finite abelian groups« (Rédei, 1967) and it has been shown that there is a logical duality between it and another important result in group theory, known as Frobenius-Stickenberger’s Theorem. Hajós’ Theorem can be formulated in an equivalent way by using the concept of periodic subset of a groupe, i.e. a subset $A$ such that there exists an element $g$ in G (other than the identity element) such that $gA = A$ . It is not difficult to see that Hajós’ main Theorem is equivalent to the statement that an integer $i$ exists such that $Ai$ is periodic. An expression in which, as I have shown before, a group $G$ has been factored in a direct sum (or, equivalently, using multiplicative notation, in a direct product of $k$ subsets) is often called a $k$ -Hajós factorisation. I will use the expression »Hajós factorisation« in the case where $k$ =2, which is the most interesting aspect of the Theory that we will discuss.
A group $G$ is said to possess the $k$ -Hajós property ¹⁴

¹⁴: In Stein and Szabó (1994) different terminology has been used!

(or to be a $k$ -Hajós group) if in every $k$ -Hajós factorisation at least one factor is periodic. In a similar way to that which was noted before, a Hajós group is a group with the 2-Hajós property. Historically the first research related to the Hajós property were attempts to study the Hajós abelian finite groups. However the distinction between Hajós groups and groups with the $k$ -Hajós property ( $k > 2$ ) is very important, for there are results which apply to the case $k > 2$ only, as shown in Fuchs (1964). Whilst it is true that results of musical interest have only been obtained, for the moment, for $k = 2$ , I will summarise all the more general results, because I think that this could help us to see new interesting applications in the musical domain.

3 List of Hajós groups

Firstly we give the complete list of groups which have the Hajós-property. We also provide a short account for the case of $k$ -Hajós groups $G$ with some particular assumption for what concerns the cardinality of every factor.
List of Hajós groups:

Finite abelian groups (see Sands, 1962).
Let $(a1,a2,...,an)$ be the short notation for the direct product $/ / / Za1 ×Za2 × ...×Zan$ and let $p,q,r,s$ be distinct primes, $a > 0$ integer.
The finite abelian groups are then all (and only) these groups (and all subgroups of them):
$a 2 2 (p2 ,q) (p ,q ) (p3,q,r) (p ,q2,r) (p,2q,r,s) (p ,2,2) (p,2 ,2) (p ,2,2,2) (p,2,22,2,2) (p,q,a2,2) (p,23,32) (3 ,3) (2 ,2) (2 ,2 ) (p,p)$
In particular it follows that cyclic finite Hajós groups are all $/ Zn$ , with $a a 2 2 2 2 n (- ,\ = {p ,p q,pq,p q,p q ,p qr,pqr,pqrs}$ and where $p,q,r,s$ are distinct prime numbers, $a > 0$ integer.