A factorization of a finite abelian group is a collection of subsets
of such that each element can be represented in the form . In this case, we write and if each contains the identity element of , we say we have a normalized factori- zation of .
The notion of factorization of abelian groups arose when G. Hajós  found the answer to “Minkowski’s conjecture” about lattice tiling of by unit cubes or clusters of unit cubes. The geometric version of “Minkowski’s conjecture” can be explained as follows:
A lattice tiling of is a collection of subsets of such that and , if , . Two unit cubes are called twins if they share a complete -dimensional face. Minkowski was wondering if there exists a tiling of by unit cubes such that there are no twins! Minkowski’s conjecture is usually expressed as follows:
Each lattice tiling of by unit cubes contains twins.
As mentioned above, it was G. Hajós  who solved Minkowski’ conjecture. His answer was in the affirmative, after translating the conjecture into an equivalent conjecture about finite abelian groups. Its group―theoretic equivalence reads as follows:
“If is a finite abelian group and is a normalized factorization of , where each of the subsets is of the form , where ; here denotes order of , then at least one of the subsets is a subgroup of ”.
Rėdei  generalized Hajos’s theorem to read as follows:
“If is a finite abelian group and is a normalized factori- zation of , where each of the subsets contains a prime number of elements, then at least one of the subsets is a subgroup of ”.
A tiling is a special case of normalized factorization in which there are only two subsets, say and of a finite abelian groups , such that is a factorization of .
A tiling of a finite abelian group is called a full-rank tiling if implies that , where denotes the subgroup generated by . In this case, and are called full-rank factors of . Otherwise, it is called a non-full-rank tiling of . As suggested by M. Dinitz  and also in that of O. Fraser and B. Gordon  , finding answers to certain questions is sometimes easier in one context than in others. In this connection consider the group, viewed as a vector space of -tuples over . Then subspaces correspond to subgroups. Moreover, is equipped with a metric, called Hamming distance , which is defined as follows:
For and ,
With respect to this metric, the sphere with center at and radius is the set .
A perfect error-correcting code is a subset of such that
and , if .
Observe that in the language of tiling, this says that is a factorization of  .
Factorization and Partition
Let be a factorization of a finite Abelian group . Then the sets
form a partition of . Also, , where as before denotes the number of elements of .
Let and be subsets of . We say that is replaceable by , if whenever is a factorization of , then so is .
Redei  showed that if is a factorization of , where
, and is a prime, then is replaceable by , for each .
A subset of is periodic, if there exists , such that
. It is easy to see that if is periodic, then , where is a proper subgroup of  .
Before we show the aim of this paper, we mention the following observation. If is a factorization of , then for any , and , then so is , so we may assume all factorizations are normalized.
Let and assume is a factorization of , where , then either or is a non-full-rank factor of .
Note that . We induct on .
If , then . Thus, is a non-full-rank factor of .
Let and assume the result is true for all such groups of order less than .
Let . Then in , by Rédei  , can be replace by
If , then is a subgroup of . Thus, , so is a non-full- rank factor of .
If , then from , we get the following partition of :
from which we get
Comparing with , we obtain . Thus, is periodic, from which it follows that , where is a a proper subgroup of . Now, from , we obtain the factorization of the quotient group , which is of order less than . So, by inductive assumption, either or from which it follows that either or . That is either or is a non-full-rank factor of QED.