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Jessica Anzanello

Publications and source records attributed to Jessica Anzanello.

6 recordsLinked to original sources
arXiv2026

Fixed-point-free elements in two-orbit permutation groups

Let $G$ be a two-orbit permutation group on $n > 2$ points. We show that $G$ contains either a derangement or an element of prime-power order with a unique fixed point. As a corollary, if the orbits of $G$ have length $n_1$ and $n_2$ and $\gcd(n_1, n_2-1) = \gcd(n_1-1, n_2) = 1$, then $G$ contains a derangement. The special case $n_1 = n_2$ was recently conjectured by Ellis and Harper and proved under various restrictive hypotheses. We prove our result by reducing to the case of simple groups and leveraging the classification of normal $2$-coverings of simple groups due to Bubboloni, Spiga, and Weigel.

math.GR↗
arXiv2025

Finite simple groups have many classes of prime order elements

Let $T$ be a finite non-abelian simple group. Giudici, Morgan and Praeger have shown that the order of $T$ is bounded above by a function depending on the maximum number of $\mathrm{Aut}(T)$-classes of elements of $T$ of prime-power order. In this note, we strengthen this result by showing, in particular, that prime-power can be replaced by prime.

math.GR↗
arXiv2025

On the proportion of derangements in affine classical groups

We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $\lambda=(\lambda_1, \dots, \lambda_m)$ into $m$ parts, with $\lambda_1\ge \dots \ge \lambda_m$, such that either $\lambda_1=1$ or $\lambda_{k-1}>\lambda_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton.

math.CO↗
arXiv2025

The maximal rank of a string group generated by involutions for alternating groups

A string group generated by involutions, or SGGI, is a pair $\Gamma=(G, S)$, where $G$ is a group and $S=\{\rho_0,\ldots, \rho_{r-1}\}$ is an ordered set of involutions generating $G$ and satisfying the commuting property: $$\forall i,j\in\{0,\ldots, r-1\}, \;|i-j|\ne 1\Rightarrow (\rho_i\rho_j)^2=1.$$ When $S$ is an independent set, the rank of $\Gamma$ is the cardinality of $S$. We determine an upper bound for the rank of an SGGI over the alternating group of degree $n$. Our bound is tight when $n\equiv 0,1,4\pmod 5$.

math.GR↗
arXiv2024

The Chebotarev invariant for direct products of nonabelian finite simple groups

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{\alpha(G)}+O(1)$, where $\alpha$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$.

math.GR↗