[Paper Review] Beauville $p$-groups of wild type and groups of maximal class

Gustavo A. Fernández‐Alcober, Norberto Gavioli+2 more|arXiv (Cornell University)|Jan 25, 2017

Finite Group Theory Research7 references3 citations

TL;DR

This paper classifies Beauville $p$-groups of maximal class within two large families: metabelian $p$-groups and those with a maximal subgroup of class at most 2. It establishes that such groups are Beauville if and only if $p \geq 5$ and either all non-fratricidal elements have order $p$, or the elements of order $p$ form exactly two maximal branches under specific conditions. Crucially, it proves the existence of infinitely many Beauville $p$-groups of wild type—groups where lifting Beauville structures from the quotient $G/\Phi(G)$ fails—thereby extending the known landscape of Beauville $p$-groups beyond the tame type.

ABSTRACT

Let $G$ be a Beauville finite $p$-group. If $G$ exhibits a `good behaviour' with respect to taking powers, then every lift of a Beauville structure of $G/Φ(G)$ is a Beauville structure of $G$. We say that $G$ is a Beauville $p$-group of wild type if this lifting property fails to hold. Our goal in this paper is twofold: firstly, we fully determine the Beauville groups within two large families of $p$-groups of maximal class, namely metabelian groups and groups with a maximal subgroup of class at most $2$; secondly, as a consequence of the previous result, we obtain infinitely many Beauville $p$-groups of wild type.

Motivation & Objective

• To classify Beauville $p$-groups of maximal class within two large families: metabelian $p$-groups and those with a maximal subgroup of class at most 2.
• To determine the conditions under which such groups are Beauville, particularly focusing on the distinction between tame and wild type $p$-groups.
• To establish the existence of infinitely many Beauville $p$-groups of wild type, which are groups where the lifting property of Beauville structures from $G/\Phi(G)$ fails.
• To extend previous results on Beauville $p$-groups of small order and limited families to a much broader class of $p$-groups of maximal class.

Proposed method

• The authors use the group-theoretic characterization of Beauville groups via two generating sets $S_1$ and $S_2$ such that $\Sigma(S_1) \cap \Sigma(S_2) = 1$, where $\Sigma(S)$ is the union of conjugates of cyclic subgroups generated by elements in the triple $T = \{x,y,xy\}$.
• They analyze the structure of $p$-groups of maximal class, particularly focusing on the maximal subgroup $G_1 = C_G(G'/\gamma_4(G))$, which plays a central role in the classification.
• The concept of a 'maximal branch' $B(M) = M \setminus \Phi(G)$ for a maximal subgroup $M$ is used to classify elements of order $p$ and $p^2$, especially in relation to the lifting of Beauville structures.
• The proof distinguishes between two cases: (i) all elements outside $G_1$ have order $p$, and (ii) the elements of order $p$ form exactly two maximal branches under congruence conditions on the group order $n$.
• They apply known results on $p$-groups of exponent $p^e$ and the criterion $|G^{p^{e-1}}| \geq p^2$ for $p \geq 5$, while identifying where this criterion fails.
• They use the structure of the infinite pro-$p$ group of maximal class to construct infinite families of examples, particularly for tame-type groups.

Experimental results

Research questions

• RQ1 Under what conditions is a $p$-group of maximal class with a metabelian or low-class maximal subgroup a Beauville group?
• RQ2 What distinguishes Beauville $p$-groups of wild type from those of tame type in the context of $p$-groups of maximal class?
• RQ3 Can infinitely many Beauville $p$-groups of wild type be constructed within families of $p$-groups of maximal class?
• RQ4 How does the lifting of Beauville structures from $G/\Phi(G) \cong C_p \times C_p$ fail in wild-type groups?
• RQ5 What role does the maximal subgroup $G_1 = C_G(G'/\gamma_4(G))$ play in determining the Beauville property?

Key findings

• All $p$-groups of maximal class of order at least $p^{p+1}$ that are metabelian or have a maximal subgroup of class at most 2 are Beauville if and only if $p \geq 5$ and either all elements outside $G_1$ have order $p$, or the elements of order $p$ form exactly two maximal branches with specific congruence conditions on the group order $n$.
• Groups satisfying condition (i) are all of tame type, meaning every lift of a Beauville structure from $G/\Phi(G)$ yields a Beauville structure in $G$.
• Groups satisfying condition (ii) are of wild type, meaning there exist Beauville structures in $G/\Phi(G)$ that do not lift to Beauville structures in $G$, due to nontrivial intersections of $p$-th powers in the center.
• The paper constructs infinitely many Beauville $p$-groups of wild type by analyzing metabelian $p$-groups of maximal class satisfying $[G_1, G_2] = G_{n-p+2}$, particularly when $n \not\equiv 2 \pmod{p-1}$.
• Every proper quotient of a $p$-group of maximal class with $p \geq 5$ is a Beauville group of tame type, showing that the Beauville property is preserved downward in the quotient lattice.
• The existence of an infinite pro-$p$ group of maximal class with all non-abelian elements of order $p$ yields infinitely many tame-type Beauville $p$-groups via finite quotients.