Index, sub-index and sub-factor of groups with interactions to number theory

Abstract

<jats:p> This paper is the first step of a new topic about groups which has close relations and applications to number theory. Considering the factorization of a group into a direct product of two subsets, and since every subgroup is a left and right factor, we observed that the index conception can be generalized for a class of factors. But, thereafter, we found that every subset [Formula: see text] of a group [Formula: see text] has four related sub-indexes: right, left, upper and lower sub-indexes [Formula: see text], [Formula: see text] which agree with the conception index of subgroups, and all of them are equal if [Formula: see text] is a subgroup or normal sub-semigroup of [Formula: see text]. As a result of the topic, we introduce some equivalent conditions to a famous conjecture for prime numbers (“every even number is the difference of two primes”) that one of them is: the prime numbers set is index stable (i.e. all of its sub-indexes are equal) in integers and [Formula: see text]. Index stable groups (i.e. those whose subsets are all index stable) are a challenging subject of the topic with several results and ideas. Regarding the extension of the theory, we give some methods for evaluation of sub-indexes, by using the left and right differences of subsets. At last, we pose many open problems, questions, a proposal for additive number theory, and show some future directions of researches and projects for the theory. </jats:p>

Subindices and subfactors of finite groups

“…right) factor relative to its right (resp. left) transversal, hence it is a two sided factor of G. It is clear that there is a right factor of G relative to A ⊆ G if and only if A is a left factor of G. In reference [4], the first author achieved a generalization of factors that not only does not have the deficiency of factors but also leads to the important concept of sub-indices for all subsets of groups.…”

“…There are some basic properties of sub-indices in arbitrary and finite groups (see [4,5]). The followings are some important results for the finite case: (a)…”

“…In the theory of sub-indices, we observe that the cardinality of subsets plays an important role for index stability. Hence we recall a definition from [4,5]. Definition 2.6.…”

“…After that, sub-factors of groups are introduced and it is shown that the concept of index of subgroups can be extended to factors and even arbitrary subsets! ( [4]). In [5] more studies for sub-indices and sub-factors of finite groups have been done.…”

“…Here, the pervious results together some new computational methods enable us to prove a main theorem that completely characterizes finite index stable groups. Also, we solve many problems, answer several questions, and prove some related conjectures by using the theoretical backgrounds form [4,5] together with the computational methods.…”