Open Journal of Discrete Mathematics
Vol.07 No.02(2017), Article ID:75432,3 pages
10.4236/ojdm.2017.72005
Non-Full Rank Factorization of Finite Abelian Groups
Khalid Amin
Department of Mathematics, College of Science, University of Bahrain, Bahrain, Kingdom of Bahrain
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: November 23, 2016; Accepted: April 14, 2017; Published: April 17, 2017
ABSTRACT
Tilings of -groups are closely associated with error-correcting codes. In [1] , M. Dinitz, attempting to generalize full-rank tilings of to arbitrary finite abelian groups, was able to show that if , then admits full-rank tiling and left the case , as an open question. The result proved in this paper the settles of the question for the case .
Keywords:
Factorization of Abelian Groups, Error-Correcting Codes
1. Introduction
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 [3] 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 [3] 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 [4] 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 ”.
2. Preliminaries
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 [1] and also in that of O. Fraser and B. Gordon [2] , 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 [6] .
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 .
Definition
Let and be subsets of . We say that is replaceable by , if whenever is a factorization of , then so is .
Redei [4] showed that if is a factorization of , where
, and is a prime, then is replaceable by , for each .
Definition
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 [5] .
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.
Theorem
Let and assume is a factorization of , where , then either or is a non-full-rank factor of .
Proof:
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 [4] , 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.
Cite this paper
Amin, K. (2017) Non-Full Rank Factorization of Finite Abelian Groups. Open Journal of Discrete Mathematics, 7, 51-53. https://doi.org/10.4236/ojdm.2017.72005
References
- 1. Dinitz, M. (2006) Full Rank Tilings of Finite Abelian Groups, SIAM Journal on Discrete Mathematics, 20, 160-170. https://doi.org/10.1137/S0895480104445794
- 2. Fraser, O. and Gordon, B. (1977) Solution to a Problem by A.D. Sands. Glasgow Mathematical Journal, 20, 115-117.
- 3. Hajós, G. (1949) Sur la factorisation des groupe abeliens, Casopis Pest. Ma. Fys., 74, 157-162.
- 4. Rédei, L. (1965) Die neue Theorie der endlihen abelschen und verallgemeinerung des hauptsatze von Hajos. Acta Mathematica Hungarica, 16, 329-373. https://doi.org/10.1007/BF01904843
- 5. Sands, A.D. and Szabo, S. (1991) Factorization of Periodic Subset. Acta Mathematica Hungarica, 57, 159-1167. https://doi.org/10.1007/BF01903814
- 6. Szabo, S. (2004) Topics in Factorization of Abelian Groups, Birkhauser, Beijing.