The hajos n-property for finite p-groups

Abstract

Abstract : Let G be a finite abelian group. Let A1 , ... A; , ... An , be subset of G; then the product A1 ... A ... An, is defined to be the set of all elements a ... a; .. . an, with a ŒA, for 1 si's n. A1 ... A . .. An may contain the same element many times. If A1 ... A; ... An contains each element only once and if every element of G is in A1 ... A; ... An, then write A1 . .. A . . . An=G and call this a factorization of G. A subset A of G is said to be periodic if there exists a nonidentity element g of G such that gA - A. A group G is said to have the Hajos n-property if in every factorization A, ... A; ... An=G, at least one of the factors is periodic. In this thesis, a study of finite p-groups with respect to the Hajos n-property is carried out. It is shown that a finite p- group does not have the Hajos n-property, if p ≥ 5, except when G is cyclic The cases of 2 and 3-groups do not show a similar pattern, and is more sporadic. However, some partial results are obtained. For example, it is shown that any group of type (2a, 2) has the Hajos n-property, for all n, while groups of type (23, 22) does not. Groups of order 34 do not have the Hajos 2-property. However, they have the Hajos 3-property.