Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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When $G$ is amenable is true that for a set $A \in p$ which is left piecewise syndetic then $\{ x: Ax^{-1} \in p \}$ is both sided syndetic?
Let be $G$ be a discrete group. I recall the definition of syndetic sets, thick sets and piecewise syndetic sets.
Definitions:
A set $A$ is left syndetic if there exist a finite $H \subset G$ such ...
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How might someone with a background in group theory start research into topos theory?
The Question:
How might an early career mathematician with a background of research in group theory start research into topos theory?
I want links between the two areas, not career advice, though it ...
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Up to what order have finite groups been classified? [duplicate]
All finite simple groups have been classified, and the classification of finite groups is thought to be wild. So, up to what order have finite groups been classified? Wikipedia tells us that it is ...
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What can we say about the average order of group members of alternating group?
During an ongoing research I dealt with the concept of orders of group members. The following question remained a gap in my analysis. Any insight is appreciated.
Let $ \bar{o}(G) $ be the average ...
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Approximating open subset of profinite group by union of cosets of ideal
I am trying to understand the proof of Theorem 1.3 in this paper by poonen. Poonen refers to Lemma 20 in a different paper. He claims that the open subset $U_P \subseteq \hat{\mathcal{O}}_P$ can be ...
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Residual finiteness of semidirect product $\mathbb{Z}^2\ltimes \mathbb{Z}[1/10]$ of abelian groups
Let $\mathbb{Z}[1/10]$ be an abelian group by addition. Let $\mathbb{Z}^2$ act on it by automorphisms by $x\mapsto 2x$ and $x\mapsto 5x$. Is the corresponding semidirect product $\mathbb{Z}^2\ltimes \...
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Lifting of algebraic fibrations via a subnormal series
Consider a finitely generated group $G$ and a subnormal series of $G$:
$$1=G_0\trianglelefteq G_1\trianglelefteq\cdots\trianglelefteq G_{n-1}\trianglelefteq G_n=G$$
Now, suppose that $G_1$ fibres, i.e....
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The geometric models of generalised Baumslag-Solitar groups
I am trying to understand a construction in the paper "The large scale geometry of the higher Baumslag-Solitar groups", GAFA, Geometric and functional analysis 11, 1327–1343 (2001), ...
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Distances on spheres in Cayley graphs of non-amenable groups
Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...
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Research directions related to the Hilbert-Smith conjecture
The Hilbert-Smith Conjecture (HSC) is a famous open problem in geometric group theory stating "for every prime $p$ there are no faithful continuous action of the $p$-adic group of integers $A_p$ ...
- 139
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Order of elements in amalgamated free products
Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
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Invariants of primary groups
In Kaplansky's book "Infinite Abelian Groups", an abelian group $G$ is called primary if every element has order power of $p$ for some fixed prime number $p$. It is well-known that every ...
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Equivalence of dihedral and symmetric group actions on a specialized real algebra
Edit: fixed misaligned indentation for "Update x and y by", below. I also had two little ideas that might help.
consider first the case where the digit 7 is not allowed, simplifying the ...
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The relation between two characteristic subgroups in finite p-group
Suppose $G$ is a finite $p$-group. Let
\begin{align*}
\mho_{1}(G)=\langle a^p\mid a\in G\rangle,\quad\Omega_{1}(G)=\langle a\in G\mid a^p=1\rangle.
\end{align*}
There are examples such that $|G|\leq |\...
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A cellular automaton with an image that is not closed
Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
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Weyl groups are Coxeter groups proof
I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...
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Examples of a group with infinitely many ends which are not represented as a free product of groups
Let $F_1$ and $F_2$-non-trivial groups.
Is it correct that the number of ends of the free product $F_1\ast F_2$ is infinite?
My thoughts about this: Since $e(G)=\infty$ then $G=F_1\ast F_2$, a non-...
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Reflections of an apartment in building — Weyl groups
I will give the definition of what I mean by reflection which is in Suzuki's group theory I.
Let $\Sigma $ be an apartment of a building that contains adjacent chambers $C$ and $C'$. Then there are ...
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Name of the power of the exponent of a $p$-group
Is there a name for the power of the exponent of a $p$-group? So, if $\mathrm{exp}(G):=\max\lbrace o(g)|g\in G\rbrace=p^k$ for some $k\in\mathbb{N}$, is there a name for the $k$? Additionally, is ...
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Can we say anything about the discrete logarithm of $x+1$?
Consider the multiplicative group $\mathbb{Z} / p\mathbb{Z}$. Let $g$ be a generator and suppose $g^n = x$. Can we say anything at all about the discrete logarithm of $x+1$? That is, can we write $m$ ...
- 591
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$G\cdot H$ with $G,H$ non-Abelian finite simple
Can a non-split extension of one non-Abelian finite simple group by another exist?
- 2,773
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Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
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Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
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Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
- 3,001
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Comparing the perfect groups of order 1344
Take two nonisomorphic perfect groups of order 1344 and label the elements of each with the numbers 1 through 1344, then superimpose their respective Cayley tables (for simplicity’s sake, the nth row ...
- 2,773
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Existence of maximal topologically characteristic subgroup of infinite index of pro-$p$ groups
Let $G$ be a topologically finitely generated infinite pro-$p$ group. Suppose that $G$ is not just-infinite. Does the group $G$ always have a maximal topologically characteristic subgroup of infinite ...
- 667
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Group theory - buildings/complexes, retractions Suzuki
I’m reading this book called group theory I by Michio Suzuki and part of a proof I just cannot figure out. It’s on page 322 proposition 3.18. I will include it here:
(3.18) Let $\Delta$ be a building,...
- 139
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Reading material similar to ‘geometry of linear groups’ chapter of Suzuki ‘Group theory I’
I am trying to find sources and other texts that cover similar material as in the title in Suzuki’s ‘group theory I’. This involves, definition and properties of buildings, complexes, the Weyl group ...
- 139
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The largest value of $k$ for $\mathbb{Z}^k$ to be embedded in $GL(n,\mathbb{Z})$
This is just a question originated from This conversation (commented by Moishe Kohan). I tried to prove those two assertions but I don't know where to start:
If H is a free abelian subgroup of $SL(n, ...
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A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$
Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
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Classification of closures of additive subgroups of $\mathbb{R}^n$
If $G$ is an additive subgroup of the real numbers $\mathbb{R}$ and $\overline{G}$ is the topological closure of $G$ then either
$\overline{G} = a \cdot \mathbb{Z}$ for some $a \in \mathbb{R}$, or
$\...
- 2,221
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Terminology for discrete subgroups of PSL(2,k), where k is a non-archimedean local field
$\DeclareMathOperator\PSL{PSL}$I'm asking about terminology for discrete subgroups of $\PSL(2,k)$, where $k$ is a non-archimedean local field.
As it is rather clumsy to have to use such expressions ...
- 586
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Arithmetic triangles and unimodality of its rows
Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence.
How to prove that the coefficients form an ...
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The largest abelian subgroups of a Lie group
Let $G$ be a semisimple Lie group. Denote $d(G)$ as the maximal integer $p$ such
that $\mathbb{Z}^p$ is isomorphic to a discrete subgroup of $G$ and $c(G)$ is the maximal integer $q$ such that $\...
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Question on a theorem about group extensions
I am reading the chapter Second cohomology groups of Continuation of the Notas de Matemàtica.
Part 1 in theorem 1.2 tells us that
Let $E : 1 \to A \xrightarrow{i} X \xrightarrow{f} G \to 1$ be an ...
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Theory of group representation for compact groups
I write here because some experts could help on that. It is very well known (at least for me) many reference books on linear representations of finite groups (for instance, the very classical and ...
- 1,561
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Projection map $\pi:\left(\mathcal{O}/n\mathcal{O}\right)^\times \to\left(\mathcal{O}/\gcd(n,m)\mathcal{O}\right)^{\times}$ of a CM elliptic curve
In this paper the author has mentioned in page $693$ under section $2.2$ that for an Elliptic curve $E/\mathbb{Q}$ with CM by an order $\mathcal{O}$ of an imaginary quadratic field $K$ there is a ...
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Commutator magma isomorphism
Define the commutator magma of a group to be the magma whose elements are the same as the group’s and whose operation is the group’s commutator.
What are the conditions for two finite groups to have ...
- 2,773
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A question about a class of pro-$\mathcal X$-group
This question concerns the following lemma of this paper:
Lemma 2. Let $\mathcal X_1,\ldots,\mathcal X_n$ be classes of finite groups closed with respect to normal subgroups and subdirect
products ...
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Is there a number field $K$ such that $K^\times / \mathbb{Q}^\times$ is finitely generated? [duplicate]
I think I have a simple proof that the only fields with finitely generated multiplicative groups are finite. What about if we take $K$ a number field and mod out $\mathbb{Q}^\times$ from its ...
- 531
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Groups of orders $7!$ and $\frac{7!}{2}$
In our research, we need to know that whether every group $G$ of order $2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7=\frac{7!}{2}$ or $5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7=7!$ has a proper subgroup non-...
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A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
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Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
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Classification of fundamental domains of a fuchsian group
Let $G$ be the (2,3,7) triangle group. We can see it as symmetry group of (2,3,7) tiling of the hyperbolic plane or symmetry group of $[3^7]$ tiling of the hyperbolic plane. This contains translations,...
- 613
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Verbally complete groups
In the summary of [Leshchenko, Yu. Yu.
The structure of an infinite wreath power construction of the regular group of prime order p. (Ukrainian. English summary) Zbl 1164.20335
Mat. Stud. 28, No. 2, ...
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An action on multiplicatively antisymmetric matrix
A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...
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Equivariant cohomology with discrete group action
As far as I know, the equivariant cohomology can be regarded as the generalisation of de Rham cohomology with group action on manifolds. From the literature, the group action is Lie group type. I am ...
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378
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Isomorphism of invariants and coinvariants over a field
Let $G$ be a finite group with normal subgroup $N$ acting on a vector space $V$ over a field $k$ in which the order of $N$ is invertible. Denote $H:=G/N$. The composite map $V^N \to V \to V_N$ and $\...
- 617
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163
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Presentation complex of a finite perfect group and its features
Let $G$ be a finite perfect group and consider $X_G$, its presentation complex. I have the following questions:
Is there any special property of $X_G$ due to the group's perfectness?
What can we say ...
- 1,177
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347
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Minimal non-solvable groups with a special property
Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group
all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$
is abelian. If for every subgroup $H &...
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