Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,181 questions
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Galois theory timeline
A recent question on the history of Galois theory wasn't the most satisfactory. But the historical issues do seem quite attractive. They relate to innovation, and to exposition. There is a perspective ...
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A Presentation for Rubik's cube group?
Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? Thus ...
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Simple discrete subgroups of Lie groups
Upon Ian Agol's suggestion, I separated this question from the one on non-residual finiteness in
Non-residually finite matrix groups
Question. Are there infinitely generated simple discrete ...
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The field of fractions of the rational group algebra of a torsion free abelian group
Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions.
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Non-linear expanders?
Recall that a family of graphs (indexed by an infinite set, such as the primes, say) is called an expander family if there is a $\delta>0$ such that, on every graph in the family, the discrete ...
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Is there a compact group of countably infinite cardinality?
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...
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What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\...
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Groups in which all characters are rational.
The Symmetric groups $S_n$ has interesting property that all complex irreducible characters are rational (i.e. $\chi(g)\in \mathbb{Q}$ for all $\mathbb{C}$-irreducible characters $\chi$,$\forall g\in ...
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Are there any computational problems in groups that are harder than P?
There are several well known classes of groups for which the word problem, conjugacy etc. are solvable in polynomial time (hyperbolic, automatic).
Then there are several classes of groups like ...
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Probability of generation of ${\mathbb Z}^2$
What is the probability that three pairs $(a,b) $ , $(c,d) $ and $(e,f) $ of integers generate $\mathbb Z^2$? As usual the probability is the limit as $n\to \infty$ of the same probability for the $n\...
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Number of finite simple groups of given order is at most 2 - is a classification-free proof possible?
This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that:
L4(2) and L3(4) both have order 20160
O2n+1(q) and S2n(q) have the same order ...
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Are there always more conjugacy classes in the kernel of a morphism to $Z_2$ than not?
Let $G$ be a finite group and let $\phi:G\to Z_2$ be a homomorphism to the group with two elements. Is it always the case that there are more conjugacy classes in the kernel of $\phi$ than conjugacy ...
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When is $S_n \times S_m$ a subgroup of $S_p$?
I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...
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Triality of Spin(8)
Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as follows. $Spin(8)$ ...
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Highly transitive groups (without assuming the classification of finite simple groups)
What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete ...
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Multiplying all the elements in a group
Let $G = \{ g_i | i = 1, ...,n \}$ be a finite group and denote by $G!$ the multiset consisting of all the products of all different elements of $G$ in any order, that is
$$ G! = [ \prod_i g_{\sigma(i)...
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Counting the Groups of Order n Weighted by 1/|Aut(G)|
Idle question:
Let $g(n)$ be the sum, over all isomorphism classes of groups of order $n$, of $\frac{1}{|Aut(G)|}$ where $G$ is a group in the class. Thus $g(n)n!$ is the number of group laws on a ...
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Realizing groups as automorphism groups of graphs.
Frucht showed that every finite group is the automorphism group of a finite graph. The paper is here.
The argument basically is that a group is the automorphism group of its (colored) Cayley graph
...
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Does GL_n(Z) have a noetherian group ring?
Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study "...
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Integer matrices which are not a power
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
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Classifying Space of a Group Extension
Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example:
$$
0 \to H \to G \to G/H \to 0\ .
$$
I want to understand the classifying space of $G$. Since ...
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Invertible matrices satisfying $[x,y,y]=x$
I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope.
Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,...
user6976
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$A^2$ is isomorphic to $A^{(\omega)}$, but not $A$
Is there an abelian group $A$ with $A\not\cong A\oplus A\cong A\oplus A\oplus A\oplus\cdots$ (a direct sum of countably many copies of $A$)?
Edited to add: As no answers are forthcoming, does anyone ...
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Have finite doubly transitive groups been classified?
I am trying to determine whether the literature contains a complete proof of the classification of finite 2-transitive groups. This is a fundamental result with important applications in many areas ...
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Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
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How large can the smallest generating set of a group $G$ of order $n$ be?
Let $n$ be a natural number. For every group $G$ of order $n$, denote
$d(G)$ : The number of elements of the smallest generating set of $G$
How large is the maximum possible value of $d(G)$ ...
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Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
Let $R$ be a finitely generated ring with identity, $M_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$? This should be an elementary ...
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About unpublished lecture notes of Philip Hall
When I study group theory, I find that there are some mysterious things. For example, Daniel Gorenstein, in his paper On a Theorem of Philip Hall, mentioned the unpublished lecture notes of Philip ...
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An anecdote by R. Schmidt
Did anybody here ever read those lines by R. Schmidt (?) where he talked about the terseness of articles in group theory in the days prior to the conclusion of the classification of the finite simple ...
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Has anyone catalogued the "first generation" proof of the classification of finite simple groups?
It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this ...
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A question about subsets of plane
Is there a subset $X$ of plane with two points $x, y$ such that each one of $X \setminus \{x\}$, $X \setminus \{y\}$ is isometric to $X$? I tried hard to construct a counterexample but failed.
Sorry ...
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"Lie algebra" for a general group ?
Is there analog of Lie algebra for the case of topological groups which are not necessarily differentiable manifolds, and in particular for finite groups? here by "analog" i mean that it should have ...
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Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
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Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of length $<\...
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Units in the group ring over fours group after Gardam
Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$.
This conjecture asserted that if $\Gamma$ is a ...
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On the size of balls in Cayley graphs
Next semester I will be teaching an introductory course on geometric group theory and there is a basic question that I do not know the answer to. Let $G$ be a finitely generated group with finite ...
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Are there two groups which are categorically Morita equivalent but only one of which is simple
Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) ...
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The Higman group
The group of Higman:
$
\langle
\
a_0, a_1, a_2, a_3 \ | \
a_0 a_1 a_0^{-1}=a_1^2,
\ a_1 a_2 a_1^{-1}=a_2^2,
\ a_2 a_3 a_2^{-1}=a_3^2,
\ a_3 a_0 a_3^{-1}=a_0^2
\ \rangle .
$
Is it simple? What is ...
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Serre's remark on group algebras and related questions
I've recently heard about an idea of Serre that for each finite group $G$ there exists a group scheme $X$ such that for each field $K$ the group $X(K)$ is naturally isomorphic to the unit group of $K[...
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How can classifying irreducible representations be a "wild" problem?
Let $q$ be a prime power and $U_n(\mathbb{F}_q)$ be the group of unitriangular $n\times n$-matrices. I've read and heard in several places (see e.g. this mathoverflow question) that classifying ...
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Does every finitely presentable group have a presentation that simultaneously minimizes the number of generators and number of relators?
This should probably be an easy question, but I don't know how to answer it: Suppose G is a finitely generated presentable group. Suppose a is the absolute minimum of the sizes of all generating sets ...
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Is there a flat manifold with trivial first homology?
Is there a closed flat manifold whose fundamental group has trivial abelianization?
The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
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Understanding "infinite" relations in groups
Consider the matrices $A = \frac{1}{5}\begin{pmatrix}5&0&0\\\ 2&2&1\\\ 2&1&2\end{pmatrix}$, $B = \frac{1}{5}\begin{pmatrix}2&2&1\\\ 0&5&0\\\ 1&2&2\end{...
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Subdirect product of perfect groups
Let $G$ be a subdirect product of finitely many perfect groups $S_1,\ldots,S_n$. That is, $G \le S_1 \times \cdots \times S_n$, and $G$ projects onto each of the direct factors $S_i$.
It is not ...
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Automatic groups - recent progress
Epstein's (et al.) "Word Processing in Groups" is a quite comprehensive monograph on automatic groups, finite automata in geometric group theory, specific examples like braid groups, fundamental ...
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Monstrous Moonshine for Thompson group $Th$?
I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
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Strong group ring isomorphisms
Background/Motivation
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
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Nonabelian topological fundamental group of a conjugate variety
Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$.
Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...
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A question on simultaneous conjugation of permutations
Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the ...
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Additive Subgroups of the Reals.
Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties ...
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