We prove a Hall $\sigma$-subgroup analogue of the McKay conjecture for $\pi$-separable groups proposed by G. Navarro. This result simultaneously generalizes the classical McKay conjecture and its $\pi$-separable version. More precisely, if $G$ is $\pi$-separable, $p$ is a prime and $\sigma=\pi\cup\{p\}$, then it is known that there exists $H$ a Hall $\sigma$-subgroup of $G$. In this case, we prove that $|{\rm Irr}_{\sigma'}(G)|=|{\rm Irr}_{\sigma'}(\mathbf{N}_{G}(H))|$.
Let $\sigma=\{{\sigma_i|i\in I}\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A group is said to be \textit{$\sigma$-primary} if it is a finite $\sigma_i$-group for some $i$.A subgroup $H$ of $G$ is said to be: \textit{$\sigma$-subnormal} in $G$ if there exists a subgroup chain $H=H...
M. T. Hussain· Hacettepe Journal of Mathema...· 0 citations
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ act...
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)...
Let $K$ be a number field and $S$ a finite set of non-archimedean places. Write $\mathcal{O}_S$ for the ring of $S$-integers of $K$ and $\mathcal{O}_S^\times$ for its unit group. Let $\pi : X \rightarrow \mathbb{P}^1$ be a morphism of (irreducible) curves defined over $K$, and denote by $\operatorname{Red}(\pi)$ the se...
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...
Let $k=\overline{\mathbb{F}_2}$. We construct a nine-dimensional $F$-rational affine variety $X$ admitting a locally closed embedding $X\hookrightarrow \mathbb{P}_k^{19}$ together with a dense open set $U\subseteq (\mathbb{P}^{19})^\vee$ such that $X\cap H$ is not $F$-rational (or even $F$-injective) for all $H\in U$....
Thomas Polstra, Austyn Simpson· 0 citations
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