Abstract
Recently sub-factors, sub-indices and infinite difference of group subsets have been introduced as a research topic by the author. We assign four sub-indices and the -difference length to every subset A of a group G. We note that all sub-indices are equal (i.e., A is index stable in G) if A is a subgroup or a normal sub-semigroup of G. It is proved that index stability of primes in integers is equivalent to a well-known open problem in number theory. In order to develop the theory, in this paper, we study sub-index properties in finite groups and give some criteria for the index stability. Also, an algorithm for finding sub-factors of finite groups is obtained. As an application of the topic, we provide some lower bounds for the order of the subgroup generated by a given subset of finite groups. Finally, we state many questions, problems, conjectures and some future directions of the topic.
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