Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.
Let $G$ be a finite abelian group, and let $c_0(G)$ be the least integer such that every regular sequence over $G$ of at least this length is an additive basis. Let $\mathsf N(G)$ be the least integer $l$ such that every weak-regular sequence $S$ over $G$ with $|S|\ge l$ has a nonempty zero-sum subsequence $T$ satisfyi...
For a nontrivial finite abelian group $G$, let $\nu(G)$ be the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is...
A. Geroldinger, Guo-Qing Wang, Wenkai Yang· 1 citation· ⚡1
Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construc...
For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\...
Let $F$ be the free group of rank two and, for a prime $p$, let $\mathbf H_p=\mathbf G_p*\mathbf{Ab}_{p-1}$ be the pseudovariety of finite groups having a normal $p$-subgroup with abelian quotient of exponent dividing $p-1$. For $H$ of rank two with $\ab_F(H)=F^{\ab}$ we determine the set $\mathfrak D(H)$ of primes $p$...
For a finite group $G$, let $\rho^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $\sigma^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $\rho$--$\sigma$ problem, we prove that \[ |\rho^{*}(G)|<2\sigma^{*}(G)...
Yong Yang· 0 citations
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