Termination of the Lattice-Automorphism Tower for Direct Products of Symmetric Groups

Abstract

Let $G$ be a finite group. Let $\mathcal{N}(G)$ be the lattice of normal subgroups ordered by inclusion, regarded as an abstract lattice. Define $\operatorname{LatAut}(G) := \operatorname{Aut}(\mathcal{N}(G))$. The \emph{LatAut tower} is the sequence defined by $G_0 = G$, $G_{n+1} = \operatorname{LatAut}(G_n)$. Let $G$ be a \emph{tower group} if $G \cong \prod_{k \geq 3} S_k^{a_k}$ with finitely many $a_k \neq 0$. We establish the following for tower groups. \emph{Product Formula.} $\operatorname{LatAut}\!\bigl(\prod_{k \geq 3} S_k^{a_k}\bigr) \cong S_{a_4} \times S_B$, where $B = \sum_{k \geq 3,\, k \neq 4} a_k$. \emph{Termination Theorem.} For every tower group $G_0$, we prove that $G_3 = 1$, and that this bound is sharp. The proof applies Goursat's lemma to classify $\mathcal{N}(G)$ into three families parameterised by admissible triples $(J,\mathbf{P},H)$ as sub-products, sign-parity elements, and mixed elements, and uses the Krull--Schmidt theorem to identify the direct factors $S_k^{(k,i)}$ as precisely the nontrivial indecomposable complemented elements of $\mathcal{N}(G)$ (the complemented elements being exactly the full sub-products). These results do not extend to groups outside the tower-group family.