A finite Hajós group is a finite abelian group 
with the following property. Whenever every element of 
has a unique expression 
with 
and 
, at least one of the subsets 
is periodic. Here periodic means that
a nonzero group element 
satisfies 
, or respectively 
. Sands (1962) completed the classification of finite Hajós
groups (Hafidh et al. 2026, pp. 3, 6). The factorization viewpoint arose
from Hajós's group-theoretic treatment of Minkowski's
conjecture on tilings by congruent hypercubes
whose centers form a point lattice (Hajós
1949, p. 158).
For example, 
(mod 12), so while the first factor is aperiodic, the second factor has period three.
Since this turns out to be the case for all tilings of 
, it is a Hajós group. The smallest case where
this is false is 
,
followed by 
.
For 
, the cyclic
group of group order 
is a Hajós group iff 
is of the form 
, 
,
, 
, 
, or 
, where 
, 
,
, and 
are distinct primes and 
a positive
integer. Non-Hajós cyclic groups therefore have orders 72, 108, 120, 144,
168, 180, 200, 216, ... (OEIS A102562).
In an undirected Cayley graph, perfect 1-codes are efficient dominating sets. Hafidh et
al. (2026, pp. 3-4) classified them for connected,
noncomplete Cayley graphs of Hajós groups
other than 
,
, 
, 
, 
, and 
. Their classification includes all such codes containing
the identity element, and the remaining codes
are their translates. It applies to both directed and undirected Cayley
graphs.
