International Journal of Algebra, Vol. 8, 2014, no. 7, 317 - 320 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4325 On the Hereditary Properties of Hajos Groups Khalid Amin Department of Mathematics University of Bahrain PO Box 32038 Sakhir, Kingdom of Bahrain Copyright © 2014 Khalid Amin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we show that three are many groups which do not possess the Hajos-nProperty. Notations and definitions Throughout this paper, by G is meant a finite abelian group. Elements of G are denoted by the letters a, b, c In particular, e stands for the identity element of G . Subsets of G are denoted by A, B, C, and subgroups by H , K , J . The number A is denoted by A . So, in particular, A denotes order of G . of elements of a subset The invariants of a group are given in the usual way, which we describe by an a example; a group of the type p , q is the direct product of cyclic groups of orders a p , q where p and q are primes. 318 Khalid Amin A1 ,, Ai ,, An are subsets of a group G , we write G A1 Ai An whenever every element g G can be uniquely represented as g a1a2 an where ai Ai , Moreover, we call G A1 A2 Ai An a factorization of G . In particular, if H and K are subgroups, G HK means that G is the direct product of H and K Namely, G HK . If G is a group and A G and g G , then gA denotes the set of all products ga where a A . Similarly A1 , A2 ,, Ai , An denotes the set of all products a1a2 an , where ai Ai . However, we emphasize that this notation AA1 A2 Ai will be used only if every element in A1 A2 Ai An can be expressed in just one way. Thus, G A1 Ai An . A subset A of G is called periodic if there is an element g e in A such that gA A. Such an element g is called a period of A. A group G is said to have the Hajos-n-property if from any factorization G A1 Ai An , it follows that at least one of the Ai is periodic. If Introduction A famous conjecture of H. Minkowsk [4] was first proved by G. Hajos [2]. He solved it after transforming the problem into a question about finite abelian groups. Below, we state two version of this conjecture: Version 1: Let G be a finite abelian group. If a1 , a2 ,an are elements of G and positive integers such that each element of G uniqerely x x x Expressible in the form a1 1 a2 2 an n , where 0 x1 rn 1 r1 , r2 ,rn are then ai e for some i,1 i n . n Version 2: Let G be a finite abelian group and A1 ,, Ai ,, An subsets of G such that G A1 Ai An is a factorization of G , then at least one of the subsets Ai is a subgroup of G . This theorem gave rise to the following question: On the hereditary properties of Hajos groups 319 G A1 Ai An is a factorization of G , does it follows that at least one of the Ai is periodic. In case this is true, we will say that G has the Hajos-n-Property. If Hajos gave a negative answer to the case n 2 by proving the existence of groups which do not have the Hajos-2-Property. One the other hand, some groups are known to possess the Hajos-n-Property. n For example, in [1], it is shown that the cyclic groups G of order p , where p is a prime has the Hajos-n-Property. Using a theorem of Redei [5], we can also deduce that the group of type (p, p) has the Hajos-n-Property. Here again p is a prime. We start with the following lemma. Lemma Suppose G has a subgroup H such that G p for some prime. H If H does not have the Hajos-n-Property. Then G itself does not have the Hajos-nProperty. Proof Let H A1 A2 Ai An be a factorization of H in which on one of Ai is periodic, and let G b1 H b2 H b2 H HB , where B b1 b1 bp is non-periodic. Then G has the factorization G b1 A1 b2 A1 bp A1 A2 An . Now, none of A2 ,, An Is periodic. Also, b1 A1 b2 A1 bp A1 is not periodic, for if it is periodic, then there must exist an element g b j ( i ) a1 e, b j ( i ) B, a1 A1 and j (i ) is a permutation of 1,2, , p such that g (b1 A1 b2 A1 bp A1 ) (b1 A1 b2 A1 bp A1 ). It follows that b j ( i ) a1 (b1 A1 b2 A1 b1 A1 ) (b1 A1 b2 A1 bp A1 ). Thaa is p p i 1 i 1 b j (i ) a1 A1 bi A1 . It follows that b j (i ) a1 A1 bi A1 . 320 Khalid Amin Hence b j ( i ) A1 bi A1 . Therefore bi b j ( i ) A1 BA1 , from which we get that 1 A1 is periodic and this a contradiction. An easy induction argument, leads to the following theorem. Results If G has a subgroup H which does not have the Hajos-n-Property. Then G itself does not have the Hajos-n-Property. Consequences An application of our result gives us that these group typies do not have the Hajos-nProperty. p a , p , a, 0, a 2 ( p a , q, r ), a 2 ( p a , p ), a 0, 0 References 1. K. Amin. “The Hajos Factorization of Elementary 3-Groups . Journal of Algebra 224, 241-247 (2000) 2. G. HAJ6S, Sur la factorization des groups abeliens, Casopies Pest. Mat. Fys. 74 (1949), 157-162. 3. O.H. KELLER, Uber die luckenlose Einfulling des Raumes Wurfeln,/. Reine Angew. Math. 163 (1930), 231-248. 4. H. MINKOWSKI, Diophantische Approximationem (Leipzig, 1907). 5. S. Szabo, “Groups With the Redei Property”, Le Matematice, Vol. LII (1997)Fasc. II., pp. 357-364. Received: March 15, 2014
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