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

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 $\mathrm{LatAut}(G):=\mathrm{Aut}(\mathcal{N}(G))$. The \emph{LatAut tower} is the sequence defined by $G_0=G$, $G_{n+1}=\mathrm{LatAut}(G_n)$. Let $G$ be a \emph{tower group} if $G\cong {\prod }_{k\ge 3}{S}_k^{a_k}$ with finitely many $a_k\mathrm{\ne }0$. We establish the following for tower groups. \emph{Product Formula.} $\mathrm{LatAut}\text{\hspace{0.17em}⁣}\left({\prod }_{k\ge 3}{S}_k^{a_k}\right)\cong S_{a_4}\times S_B$, where $B={\sum }_{k\ge 3,k\mathrm{\ne }4}{a}_k$. \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.