The Minimum Size of a Poset Realizing ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$ as its Automorphism Group

Abstract

For a finite group $G$, let $\beta (G)$ denote the minimum cardinality $|P|$ among finite posets $P$ whose automorphism group $\Aut(P)$ is isomorphic to $G$. While every finite group is realizable as the automorphism group of some finite poset, exact values of $\beta (G)$ are known only in special cases, most notably for cyclic groups. In this paper we prove that $\beta ({\mathbb{Z}}_2\times {\mathbb{Z}}_4)=14$; in particular, the product bound $\beta (G\times H)\le \beta (G)+\beta (H)$ is sharp in this case. The upper bound is realized by an explicit $14$-element poset $P_{14}$, whose automorphism group is computed by a height-function argument together with a rigidity analysis of its covering relations. The lower bound, which constitutes the substantive part of the proof, is established by a case analysis of the orbit decompositions of a hypothetical poset on at most $13$ points under a faithful $G$-action, organized according to the largest orbit size; in each case we construct an order-automorphism outside the given copy of $G$, contradicting $\Aut(P) \cong G$. Among non-cyclic groups, to our knowledge this is the first exact determination of $\beta (G)$ whose lower bound requires a structural analysis of this kind: for the other non-cyclic abelian groups of order at most $8$, namely ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$, the value of $\beta$ is elementary. The arguments are closely adapted to the subgroup lattice of ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$.