Generation of Generalised Wreath Products of Symmetric Groups

Abstract: Let I be a finite partially ordered set and let (Sym(Δi),Δi)i be a sequence of symmetric groups indexed by I. Construct the generalised wreath product (F, Δ) on this sequence of permutation groups. We determine the minimum number d(F) of generators required for this generalised wreath product.

• The paper establishes that the minimum generator count for the generalized wreath product F equals the size |I| of the indexing poset for |I| ≥ 2.
• It employs recursive semidirect and wreath product decompositions to prove the result, constructing explicit generating sets while leveraging key symmetric group properties.
• The findings provide crucial insights into permutation group generation that impact algorithmic group theory, combinatorial designs, and structured symmetries.

Summary and Main Result

The paper "Generation of Generalised Wreath Products of Symmetric Groups" (2604.16141) explores the minimum generator count for generalized wreath products formed from symmetric groups indexed by finite posets. The principal theorem states that given a finite partially ordered set $I$ with $|I|\geq 2$, and non-trivial symmetric groups $Sym(\Delta_i)$ acting on sets $\Delta_i$ for each $i\in I$, the generalized wreath product $F$ formed from these groups has minimum generator number $d(F) = |I|$.

This result holds for arbitrary posets, emphasizing its generality. The proof leverages recursive semidirect and wreath product decompositions, and the structure theory of permutation groups indexed by poset block structures.

Background and Motivation

The minimum number of generators of a finite group is a classical topic, with symmetric groups and non-abelian simple groups known to be generated by two elements. Prior works by Wiegold, Quick, Bondarenko, and others established generator growth rates in direct and iterated wreath products, and characterized conditions under which products are finitely generated.

Generalized wreath products, as defined by Bailey et al. [Generalised], encompass direct products, conventional wreath products, and their compositions, determined by the structure of $I$. When $I$ is an antichain, $F$ is a direct product; when $|I|\geq 2$0 is a chain, $|I|\geq 2$1 is an iterated wreath product. These constructions are linked to poset block structures and their automorphism groups.

Generalized Wreath Product Construction

For a finite poset $|I|\geq 2$2, the construction involves sets $|I|\geq 2$3 (all non-singleton) and permutation groups $|I|\geq 2$4 (here, necessarily symmetric groups) acting on $|I|\geq 2$5. The generalized wreath product $|I|\geq 2$6 is realized as a group of functions $|I|\geq 2$7 from $|I|\geq 2$8 (where $|I|\geq 2$9 is the ancestral set above $Sym(\Delta_i)$0) to $Sym(\Delta_i)$1, with $Sym(\Delta_i)$2. The group operation is defined via composition involving projections and ancestral set structure, ensuring group status and compatibility with poset indexing.

Key structural features include:

• Subgroups tied to minimal elements ($Sym(\Delta_i)$3), normality criteria, and semidirect decompositions.
• Decomposition results: for minimal $Sym(\Delta_i)$4, $Sym(\Delta_i)$5 for $Sym(\Delta_i)$6. If $Sym(\Delta_i)$7 is incomparable, the product is direct: $Sym(\Delta_i)$8.
• Specific generator sets ($Sym(\Delta_i)$9) constructed so that $\Delta_i$0 when each $\Delta_i$1 is transitive.

The structure adapts to poset type:

• Chains correspond to iterated wreath products (Lemma~\ref{lem:chain}).
• Antichains result in direct products (Lemma~\ref{lem:antichain}).

Proof Structure and Numerical Results

The proof of the main theorem proceeds by induction on $\Delta_i$2. For $\Delta_i$3 or $\Delta_i$4, all possible poset structures are enumerated, and in each case, explicit generator counts are calculated using decompositions and prior known results (e.g., for regular wreath products [LuQuick], direct products, and cases corresponding to specific poset configurations).

Key numerical assertions include:

• For symmetric groups and poset antichain, the minimum generator count equals the number of factors.
• For iterated (chain) wreath products, results from [LuQuick] confirm $\Delta_i$5.
• For more complex posets (e.g., the "triangle" and "pyramid" cases for three elements), explicit generating sets are constructed, and the minimum count shown to be $\Delta_i$6.
• Every generalized wreath product of symmetric groups indexed by a poset has a quotient isomorphic to $\Delta_i$7, giving $\Delta_i$8.

The inductive step relies on semidirect and wreath product decompositions, leveraging properties of symmetric groups and the action types induced by the poset structure. The upper bound is achieved by explicit construction of generators, while the lower bound is obtained via the homomorphic image argument.

Structural and Theoretical Implications

This result provides a uniform characterization of generator requirements for a broad class of permutation groups, including those arising from automorphism and block stabilizer settings tied to poset block structures. The explicit dependence of $\Delta_i$9 on $i\in I$0 is both natural and optimal, reflecting the irreducible complexity imposed by the indexing poset.

Beyond group generation, the analysis allows concrete calculation of generator complexity for groups acting on structured combinatorial objects, which is significant for algorithmic group theory, classification of permutation groups, and combinatorial designs.

Related works on crown-based products, probabilistic generation, and block structure automorphisms are unified within this framework by virtue of the generalized wreath construction.

Future Directions and Applications

Potential extensions include:

• Analysis for other group types and action types, such as alternating groups, cyclic groups, or more exotic permutation groups.
• Investigation of generation probabilities and distribution of generating sets for generalized wreath products, building on methods from [MRQ2].
• Application to algorithms for determining minimum generating sets in finite group computation, informed by poset-based decompositions and automorphism structures.
• Links to coding theory, combinatorial designs, and block structure symmetries, where group actions indexed by posets are pivotal.

Further theoretical development may explore the impact of poset structure on other group invariants, such as commutator subgroup ranks, subgroup growth, and representation-theoretic properties.

Conclusion

The paper rigorously establishes that the minimum number of generators $i\in I$1 for a generalized wreath product $i\in I$2 of symmetric groups indexed by a finite poset $i\in I$3 equals $i\in I$4. This result is achieved via structural decomposition, induction, and explicit generator construction, and connects with broader theory on permutation group generation and poset-indexed symmetries. The implications span theoretical group theory and computational applications in algebra and combinatorics.