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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 …

This is a version of this question of Klim Efremenko. Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$ be an irreducible complex representation of $G$. We con …

$\newcommand{\ssc}{\text{sc}} \newcommand{\der}{\text{der}} \DeclareMathOperator{\im}{im}$Yes, this is true for a reductive $F$-group $G$, if for the simply connected semisimple $F$-group $G^\ssc$, s …

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where eac …

Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity comp …

This seems to have nothing to do with algebraic groups. I suppose that you mean that $G$ is a connected reductive group. So you have an irreducible algebraically variety $G$ and an open subvariety $X= …

Let $G$ be a finite group. Let $M$ be a finite $G$-module (a finite abelian group with an action of $G$). We consider a special kind of $G$-modules; in particular, our $M$ is a finite dimensional repr …

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, …

Let $G$ be a finite group. By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$. We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis …

I assume that the question is: Is it true that there is always a prime $\ell$ such that $G_\ell$ has no absolutely simple direct factors? The answer is YES. There are infinitely many such $\ell$. …

Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis permut …

Let $G$ be a finite group and $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. Consider the functor $$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm T …

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 …

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. M …