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$\newcommand{\sss}{{\rm ss}} \newcommand{\ssc}{{\rm sc}} \newcommand{\tor}{{\rm tor}} \newcommand{\X}{{\sf X}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\qed}{{$\blacksquare$}} $Let $G$ be a (connected …

Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma: Lemma. Assuming that $p$ is "good" for $G$, there ex …

Here I give details of the reduction in LSpice's comment. I write it as an answer rather than a string of comments in order to have an editable text. The reduction goes as follows. According to Will's …

1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$. Note that any subgroup of $G$ contains $1_G$, and so the set of all s …

Let $G$ be a group and $M$ be a $G$-module, that is, an abelian group written additively on which $G$ acts: $$ (g,m)\mapsto g m.$$ We consider the group of coinvariants $$ M_G:=G/\langle g m -m\ |\ g\ …

Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. We work with the first homology group $$ H_1(G,M).$$ For any c …

$\newcommand{\ab}{{\rm ab}} \newcommand{\ord}{{\rm ord}} $Let $G$ be a finite or profinite group. Consider the abelianized group $$G^\ab=G/G'$$ where $G'$ is the commutator subgroup of $G$. Let $H\sub …

Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$ of cardinality $2m\ge 6$ where $m$ is odd. Question 1. Is it true that $G$ always has a subgroup $H$ of index 2 …

LSpice answered my Question 4 in the positive. Here I deduce the positive answer to Question 3, which I restate as Question 3'. Question 3'. Let $G$ be a finite group acting transitively on the finit …

Let $G$ be a finite group of order $2m$ where $m>1$ is an odd natural number. Question. Is it true that any such $G$ has a subgroup $H$ of index 2? If yes, I would be grateful for a reference or a …

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\im{im}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\SL{SL}$I add details to Friedrich's comment: The question easily reduces to the case of a sem …

This seemingly elementary question was asked in Mathematics StackExchange.com: https://math.stackexchange.com/q/4779592/37763. It got upvotes, but no answers or comments, and so I ask it here. Let $G$ …

$\newcommand{\wt}{\widetilde}$ Let $n=1,2$. There are infinite torsion abelian groups $H^1$, $H^2$ killed by some natural number $m$. There are finite subgroups $$ {\rm Sha}^1 \subset H^1,\quad …

Let $X={\rm Gr}_{n,k,{\Bbb R}}$ denote the Grassmannian of $k$-dimensional subspaces in ${\Bbb R}^n$. We regard $X$ as an ${\Bbb R}$-variety with the set of complex points $X({\Bbb C})={\rm Gr}_{n,k,{ …

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED Let $G=\SL_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the c …