Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,183 questions
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left integration of functor in the category of groups
Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?
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Central-by-cyclic
This is a following-up question of this.
Lemma 2.4 from Robert Griess' paper "Elementary abelian $p$-subgroups of algebraic groups" states:
(i) Let $T$ be a finite $p$-group whose Frattini ...
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$p$-modular splitting systems and the characteristic of the ring $\mathcal{O}$
Let $k=\overline{k}$ be a field of characteristic $p$.
Let $(K,\mathcal{O},k)$ be a $p$-modular system.
Let both $k$ and $K$ be splitting fields for $G$ and its subgroups.
The ring $\mathcal{O}$ can ...
- 311
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Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
- 356
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Finite groups of prime power order containing an abelian maximal subgroup
Let $G$ be a finite $p$-group containing an abelian maximal subgroup. Then it is a well-known result that $|G:Z(G)|=p|G'|$. If in addition $G$ is of nilpotent class 2, then $|G:Z(G)|\leq p^{r+1}$, ...
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Constructing tensor structures for representations over group schemes
Let $A$ be an algebra over a field $k$. Let's say a tensor structure for modules over $A$ is any functorial assignment of an $A$-module structure to $M\otimes_kM'$ for $A$-modules $M, M'$. A good way ...
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Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
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Totally imprimitive groups
Following [Neumann P.M., The lawlessness of groups of finitary permutation groups, Arch. Math. 55(6) 1990, 521-532], we define totally imprimitive groups in a more general form as follows:
Let $G$ be ...
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Weyl group is the set of type preserving automorphisms
So in Suzuki’s group theory I we have a proposition 3.20 (ii) which says if we have a building $ \Delta $ and $\Sigma $ an apartment with $C \in \Sigma $ a chamber. Also we have $B \in \Sigma $ is any ...
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List of automorphism groups of low-dimensional complex commutative algebras?
Let $\mathcal{A}$ be a finite-dimensional commutative associative unital $\mathbb{C}$-algebra. I am looking for a list (of further examples of) $\operatorname{Aut}_\mathbb{C}(\mathcal{A})$, the group ...
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Limit of groups with Kazhdan property (T)
Let $G_1 \le G_2 \le \cdots $ be countable groups with Kazhdan property (T). Let $G = \bigcup_i G_i$. Does it necessarily follow that $G$ has (T)?
This seems false but I cannot find a counterexample.
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Convolutions of (m)-associahedra and (m)-noncrossing partition polynomials--combinatorial proofs?
I'm looking for combinatorial proofs of the convolutional identity COP below and its specializations I) and II).
(Edit 6/2/2023: A combinatorial proof is sketched in a blog post by Mike Spivey of a ...
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Is there such a function for the class of finitely presented groups?
By a retract of a group $G$, we mean a subgroup $H$ of $G$ for which there is a homomorphism $r:G\to H$ such that $r\circ i=1_H$, where $i:H\to G$ in the inclusion. By a proper retract, I mean a ...
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A closed subgroup of $p$-adic analytic group having same dimension is open?
Let $G$ be $p$-adic analytic pro-$p$ group and $H$ a closed subgroup of $G$. Suppose that $G$ and $H$ have the same dimension as $p$-adic analytic groups.
Question: Is it true that $H$ is an open ...
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$C_G(E)= E \times{\rm PGL}_k(q)$
Let $r$ be an odd prime and $q$ a power of a prime $p$ where $r\neq p$.
If $r^m|n$ and $q\equiv1$ (mod $r$), then $r^{1+2m}.{\rm Sp}_{2m}(r)\le{\rm GL}_n(q)$ and $Center(r^{1+2m}.{\rm Sp}_{2m}(r))\le ...
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Groups without "almost equivariant" coarse embeddings
Let $X$ be a set. We say that $\psi:X\times X\to[0,\infty)$ is a CND (conditionally negative definite) kernel if there is a Hilbert space $\mathcal{H}$ and a map $f:X\to\mathcal{H}$ such that
\begin{...
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Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
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About the question "Tannaka–Krein duality"
I saw this post recently: Tannaka–Krein duality
I have this question please: in the following which I report here:
The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}...
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Units in group rings in SAGE
Is there a recorded/known SAGE code to compute units in integral group rings for finite abelian groups ?
I would be happy with a code that only works for cyclic groups. I sort of know how to ...
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Irreducible module of finite simple groups
Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$.
Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module.
I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$...
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tensor dimension/reshaping group
Consider an $N$ dimensional tensor $T$ using the strided view representation used by PyTorch, i.e. we have a storage vector $S$ projected into $N$ dimensions using a size tuple $s$ and a stride tuple $...
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Distinguish $p'$-elements in a coset
Let $G$ be a finite group with a nonabelian minimal normal subgroup $N$.
Then $N$ is a direct product of $n$ copies of some nonabelian simple group $S$.
Let $p$ be a prime divisor of $|S|$ and let $xN\...
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Trivial homology groups for p-torsion groups
Let $G$ be a group where each element has a $p$-power order.
Let $M$ be a $G$-module without $p$-torsion.
Here $G$ is a discrete infinite subgroup of a complete group. Then, it cannot be assumed pro-...
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Bottleneck edge in lattice of subgroups
Let $G$ be a finite group. Define the bottleneck weight of a chain of subgroups $$\operatorname{id}=H_0 < H_1 < \ldots < H_n = G$$ to be the maximum value over the indices $[H_{i+1} : H_i]$ ...
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Is the commutator of the holomorph of generalized quaternion group abelian?
Let $Q_{2^{n}} = \langle x, y \mathrel\vert x^{2^{n-1}}=y^4 = 1, x^{2^{n-2}}=y^2, y^{-1}xy = x^{-1} \rangle$ be the generalized quaternion group of order $2^{n}$.
Let $\operatorname{Hol}(Q_{2^{n+1}})$ ...
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A duality of finite groups coming from a surjective homomorphism with finite kernel of algebraic tori
$\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Gm}{{{\mathbb G}_{m,{\Bbb C}}}}
\newcommand{\X}{{\sf X}}
$ I am looking for a reference for the following lemma (for which I know a proof):
Lemma.
Let $\...
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Grothendieck group of finite cyclic groups
Let $\mathcal{C}$ be the set of isomorphic classes of all finite cyclic groups $[C_n]$, with $C_n \cong \mathbb{Z}/n\mathbb{Z}$, $n \in \mathbb{N}_0$.
Define relations on $\mathcal{C}$ as follows: if $...
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Representation of a group with dimension equal to a number of conjugacy classes
My question is the following: is there a (call it "canonical", "standard", or some other interesting and known) representation (probably reducible) of a finite group, which ...
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Character extension about $Q_8$
Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise:
(Exercise 5.9)
Let $G$ be a finite group and $N\unlhd G$, suppose ...
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Generators of of $p$-adic congruence subgroups of $\operatorname{SL}_2$
$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime, $i$ a fixed positive integer and let $\Gamma_i$ denote the kernel of the map $\SL_2(\mathbb{Z}_p)\rightarrow \SL_2(\mathbb{Z}_p/p^i\mathbb{Z}_p)$. ...
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Classification of the normal subgroups of the discrete Heisenberg group
Let $H$ be the discrete Heisenberg group, i.e., the set of matrices of the form
$\begin{bmatrix}
1 & x & z \\
0 & 1 & y \\
0 & 0 & 1
\end{bmatrix}$
where $x,y,z \in \mathbb{Z}$...
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Kronecker product preserves the conjugacy relation?
Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-...
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What is the kernel of the differential of the orbit-stabilizer map for nonsmooth stabilizers?
$\newcommand{\Lie}{\operatorname{Lie}}$Let $G$ be a smooth linear algebraic variety over perfect field $k$, acting on a separated variety $X$, and for $x \in X(k)$ write $G_x$ for the scheme-theoretic ...
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Presentation complexes with same homology and different fundamental groups
If we start with a perfect group $G$ of deficiency zero then there is a presentation $P$ of $G$ such that the number of relations and the number of generators for $P$ are the same. For such $P$, the ...
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Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
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Embedding (Kronecker product) preserves the structure?
In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix}
-I_{i} & 0\\
0 & I_{n-i}
\end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
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Question on the representative of the longest Weyl element of $\mathrm{SO}(2n+1)$
Let $w_{m}$ be the $m \times m$ matrix with ones on the non-principal diagonal and zeros elsewhere.
Let $V$ be the $2n+2$-dimensional quadratic space with the symmetric bilinear form $\left<,\right&...
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Centralisers of involutions not quasi-isolated
The quasi-isolated elements of the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$ are classified in the paper "quasi-isolated elements in reductive groups" by C. Bonnafe.
Let's focus ...
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Generalized words [closed]
Dan Segal, in his book 'Words', has defined generalized words. I have trouble understanding generalized words. What I have understood from the definition of generalized words are as follows:
Let $X = \...
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Is the element in the connected component?
I posted this question at stack exchange, got two upvotes but no answer. If it doesn't belong here, please let me know.
In the algebraic group $G$ = PGL$_{8}$($\mathbb{C}$), there are two involutions $...
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Finite simple groups of order $p+1$
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSU{PSU}$Cross-post from MSE. There are some very interesting comments on the original post if you want to go check it out.
Are there any well known ...
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Functional equation $f(x*y) = f(f(x)*f(y))$
Find all endo-functions $f$ on a commutative semigroup $(\mathbb{S},*)$ such that
$f(x*y) = f(f(x)*f(y))$.
Typical case of interest are $(\mathbb{N},+)$ or $(\mathbb{Z}/k\mathbb{Z},+)$ or $(\mathbb{Z}/...
- 1,306
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Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
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A weakening of infinite Golomb rulers for group actions
If a group $G$ acts on a space $X$, then a Golomb ruler is a subset $A$ of $X$ such that $|gA\cap A|\le 1$ for all $g\in G\backslash\{e\}$.
I am interested in a weaker concept, let's call it a "...
- 2,555
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Realization of a subgroup in a maximal subgroup of a classical group
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}$In the finite group $G = \operatorname{PGL}_8(5)$, there is an elementary abelian $2$-subgroup $E$ of rank 5. $E = A_{1} \times A_{2} $ where $...
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Number of ways to write a group element as a product of generators
Let $G$ be a finite group generated by some finite set $S = \{g_1, g_2, ...\} \subseteq G$. Let $h \in G$ be some element. Let the function $c_n: G \rightarrow \mathbb{N}$ be defined that $c_n(h)$ is ...
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Definition of arithmetic subgroups of Lie groups
In Maclachlan-Reid we can read
Let $G$ be a connected semisimple Lie group with trivial centre and no compact factor. Let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Then $\Gamma$ is ...
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The influence of the derived subgroup of the unit group of a group algebra
Let $FG$ be a group algebra in which $K$ is a field and $G$ is a group. Suppose that every element in the derived subgroup $\mathcal{U}(FG)'$ of the unit group $\mathcal{U}(FG)$ of $FG$ is a ...
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Associating octonionic matrices
Is it possible for three square octonionic matrices to associate multiplicatively even though some or all of their entries do not? If so, is it possible to construct matrix groups in this way?
- 2,773
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When are normal forms computable in amalgamated produts and HNN extensions
I have had little luck searching for references on the following. I would thank a lot any reference.
What are known conditions that ensure computable normal forms on amalgamated products and HNN ...
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