Embeddability Degree of Finite Groups and Approximate Embeddings

Source abstract

This study introduces the embeddability degree , a new probabilistic invariant for finite groups motivated by the idea of measuring how closely one group can be embedded into another when an exact embedding does not exist. This proposed invariant provides a quantitative way to evaluate injective maps according to how frequently they preserve the group operation. We establish several fundamental properties of this notion, including a characterization of the case where the embeddability degree is equal to one, a criterion for its vanishing, a symmetry result for groups of equal order, and a composition-type inequality involving three finite groups. We also compute the invariant explicitly for several small cyclic groups, showing that even in elementary cases it reflects meaningful arithmetic and structural features. These initial results suggest that the embeddability degree offers a natural quantitative perspective on classical embedding problems in finite group theory.