MC-finiteness of finite group structures

Determine the number of labeled ternary structures on an n-element universe that define finite groups, and establish whether the resulting counting sequence is MC-finite.

Background

The paper considers ternary structures representing binary operations, including groups and semigroups. Because the Gaifman graph of a binary operation represented as a ternary relation is complete, these classes generally have unbounded G-degree, placing them outside the bounded-degree setting in which the paper proves MC-finiteness of density functions. The authors therefore ask whether the number of labeled finite group structures satisfies the MC-finiteness property.