What question did this study set out to answer?

The research seeks to determine whether a non-empty subset of a group can be an internal edge of a rough subgroup and to identify any corresponding subgroups.

February 14, 2026Open Access

Rough approximation of subgroup and conjugacy relation

Key Points

The research seeks to determine whether a non-empty subset of a group can be an internal edge of a rough subgroup and to identify any corresponding subgroups.
Defined the concept of rough subgroup and internal edge in relation to a conjugacy relation.
Explored mathematical properties of subsets in relation to these groups.
Evaluated the conditions under which a subset can be classified as an internal edge.
Identified criteria for subsets to qualify as internal edges of rough subgroups.
Found that such internal edges can correspond to specific subgroups, which may or may not be unique.

Abstract

Let G be agroup and H be a rough subgroup of G with respect to the conjugacy relation, which is considered as an equivalence relation. An internal edge of H is defined as the difference between H and its lower approximation. Let E be a non-empty subset of G. In this paper, we aim to answer the following questions: Can E represent an internal edge of some subgroup of G (in other words, what are the conditions that E must satisfy in order to be an internal edge of some subgroup of G)? If the answer to this question is yes, what is this subgroup, and is it unique or not?

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