Generation of Iterated Wreath Products Constructed from Almost Simple Groups

Let G1, G2, … be a sequence of almost simple groups and construct a sequence (Wi) of wreath products via W1 = G1 and, for each i > 1, Wi+1 = Gi+1 wr Wi via the regular action of each Gi. We determine the minimum number d(Wi) of generators required for each wreath product in this sequence.

💡 Research Summary

The paper investigates the minimal number of generators required for iterated wreath products built from almost simple groups via their regular actions. Let (G_{1},G_{2},\dots) be a sequence of almost simple finite groups, and define the iterated wreath products recursively by (W_{1}=G_{1}) and (W_{i+1}=G_{i+1}\wr W_{i}) where each wreath product is taken with respect to the regular action of the top factor. The main goal is to compute the invariant (d(W_{i})), the smallest size of a generating set for each (W_{i}).

The authors begin by recalling the standard definition of a (permutational) wreath product and the notation (d(G)) for the minimal number of generators of a finite group (G). They review key results from Lucchini and Menegazzo, especially the theorem that for a non‑cyclic finite group with a unique minimal normal subgroup (N) one has (d(G)=\max{2,d(G/N)}). They also introduce the augmentation ideal (I_{G}) of the integral group ring and the presentation rank (\operatorname{pr}(G)=d(G)-d(I_{G})), which will be used to relate generating properties of wreath products to module‑theoretic data.

A central tool is Lucchini’s formula for the minimal number of generators of a wreath product (A\wr G) where (A) is a finite abelian group and (G) acts regularly. In Lemma 2.4(vi) Lucchini shows \