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

Infinite groups.
This case doesn’t appear to be completely solved (see Fuchs, 1964) because the restriction has been imposed such that the order of one factor is finite.
Let $/Z(p oo )$ be the Pruferian group, i.e. the abelian infinite group in which every element has finite order $a p ,a > 0$ integer. Three cases are distinguishable:

The torsion free case: $Z/$ (see Hajós, 1950) and $|Q|$ (see Sands, 1962).
The torsion case (see Sands, 1959): $oo Z/(p )+Z/q$ (i.e. direct sum), and all subgroups of these, where $p$ and $q$ are prime distinct numbers (except for the case $p = q = 2$ , which is admitted).
The mixed case:
$|Q|+Z/p$ , $Z/+Z/p$ and all subgroups of these, where $p$ is prime.

List of $k$ -Hajós groups:
Sands has studied the case of the Hajós $k$ -property for finite cyclic groups $G$ with the assumption that ”every factor has a prime power of elements” (cfr. Fuchs, 1964, p. 139). The following cyclic groups are shown to be $k$ -Hajós groups:

$Z/pn$
$Z/ n p q$
$Z/m$ , where the exponential sum $e(m)$ of $m$ is $k$ (where $m = pn11 ...pnss$ and $e(m) = n1 + ...+ ns$ , $pi$ distinct primes).

Sands shows that a cyclic group which does not have the Hajós $k$ -property $(k > 2)$ is $/Zn$ , where $a 2 n = p q$ . But this restriction can be relaxed by affirming that the previous cyclic group is not $(k - 1)$ -Hajós group for each $k > 2$ .
We now follow the metamorphosis of Vuza’s modal theory into a rhythmic domain which naturally brings us to a new perspective about some groups relevant to musical composition. In Vuza’s theory of periodic rhythmic canons (as discussed in Vuza, 1985, 1986, 1991-93), cyclic groups which do not have Hajós property are fundamental in the formalisation of particular rhythmic canons. There is apparently no limitation in the order of the cyclic group $Z/ n$ for a musically relevant application because the cyclic group itself applies to Vuza’s definition of rhythm and, more generally, of rhythmic class. As pointed out in Mazzola (1990) in considering Olivier Messiaen’s theoretical and compositional ideas about »modes of limited transposition« and »non invertible rhythms«, the analogy between the two concepts is far from »adequate«. Vuza’s new algebraic constructions, based on the notion of action of a group $G$ on an ensemble principal homogéne $S$ (or $PH$ $G$ -set, cf. Vuza (1988)), furnishes a complete analogy (i.e. isomorphism) between the