Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,094 questions
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Counting transitive generators according to coset type
Let $\sigma=(1\;2)(3\;4)\cdots (n-1\; n)$ be a fixed-point-free involution in $S_{2n}$. I want to count permutations $\pi$ such that the group $\langle \pi,\sigma\rangle$ generated by $\pi$ and $\...
- 1,549
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Systems of imprimitivity for unitary representations - reference request
Let $G$ be a finite subgroup of the group $U_d(\mathbb{C})$ of unitary transformations of $\mathbb{C}^d$. Suppose that $G$ acts irreducibly but is imprimitive, meaning that there is a nontrivial ...
- 4,776
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Graphs of groups with homomorphisms not necessarily injective
I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general ...
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What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
What is $\mathrm{Hom}(\mathbb{Q},\mathbb{Z}(p^{\infty}))$?
I have a reference that says the group in question is $\mathbb{Q}_p,$ the additive group of the quotient field of the $p$-adic integers. Can ...
- 549
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(Z/n)^(I) is a direct summand of (Z/n)^I
Dear group theorists,
Let $n \geq 1$ and $I$ be an infinite set (you may assume $I$ to be countable). Is the abelian group $(\mathbb{Z}/n)^{(I)}$ (direct sum of copies $\mathbb{Z}/n$) a direct ...
- 63.1k
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Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
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Hurwitz's automorphisms theorem with deformations
Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus.
Now suppose a finite group $G$ acts faithfully on a Riemann surface $...
- 17.6k
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Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?
Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S_n$.
Let me give "field with one element" interpretation ...
- 24.9k
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Groups (not) quasi-retracting onto $\mathbb{Z}$
Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...
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Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
- 6,180
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How large can abelian subgroups of class 2 nilpotent groups or simple groups be?
If $G$ is a finite simple group then is it true that an abelian subgroup $H$ of $G$ of maximal order has order $|H| < |G|^{\frac{1}{3}}$? If so, could you please point me to a reference for this, ...
- 181
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Projective representations of a finite abelian group
Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups
$$
G\cong ...
- 813
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Words which are not inverted by any endomorphism
Let $w$ be a word in a free group $F_2$ of two generators $x_1, x_2$ such that there does not exist any endomorphism of free group which takes $w$ to $w^{-1}$. Let $w_1, w_2$ be two words in the same ...
- 355
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Finding a basis for the (linear combinations) span of a matrix group, efficiently?
I have an algorithm whose bottleneck is the following task:
Let $\mathbb{F}$ be a finite field.
Given a set of $k$ invertible matrices $g_1,\dots,g_k\in GL_n(\mathbb{F})$, let
$G=\langle g_1,\dots,...
- 3,104
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Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
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Action of a profinite group
Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
- 11.3k
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Finitely generated solvable groups all of whose abelian normal subgroups are finite
Is there a classification for infinite finitely generated solvable groups all of whose abelian normal subgroups are finite?
I mean by classification something like presentation.
Edited: Is there an ...
- 3,738
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Structure of abelian connected complex linear algebraic groups?
Let $G$ be an abelian connected complex linear algebraic group.
Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated ...
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Can we classify all finite 2-generated groups $G$ such that if $x,y$ generate $G$, then so does $x,yxy^{-1}$?
Can we classify finite 2-generated groups $G$ satisfying the following property:
For any pair $x,y$ which generate $G$, the pair $x,yxy^{-1}$ also generates $G$.
By the comments, no nontrivial ...
- 10.7k
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Root system automorphisms as inner automorphisms of extended Chevalley group
For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
- 3,186
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Atlas of finite groups, Character table of automorphism group of sporadic group
I am consulting ATLAS of finite group for character table of Automorphism Group of sporadic group.
I am reading from Inverse Galois Theory by G. Malle
Let me start with $G=M_{12}$
This(image ...
- 591
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In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
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Non-trivial extension of representations have same central character
Let $\pi_1, \pi_2$ be two irreducible complex representations of $G=\mathrm{GL}_2(\mathbb{Q}_p)$ and assume that there exists a non-split extension $0\to\pi_1\to \pi\to\pi_2\to0$ of representations ...
user515481
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Spectral sequence construction of Euler class of group extension
Let $A$ be an abelian group equipped with an action of a group $G$ and let
$$1 \longrightarrow A \longrightarrow \Gamma \longrightarrow G \longrightarrow 1$$
be an extension of group inducing the ...
- 51
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What is God's number for the WrapSlide puzzle?
WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
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Are homotopy braid groups residually nilpotent?
A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...
- 1,108
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A functional equation for a family of functions indexed by the symmetric group $S_3$
$\newcommand{\C}{\mathbb C}$A question asked recently was as follows:
For the symmetric group $G:=S_3$, is it possible to construct functions $t_g\colon\C\to\C$ that satisfy the convolution identity
...
- 128k
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Burnside group $B(2, 3)$ has $27$ elements, isomorphic to unitringular matrix group $\text{UT}(3, 3)$?
I realized my question here might have been too hard for MSE, so I'm asking it here as well.
The Burnside group $B(d, n)$ is defined as the quotient of the free group on $d$ generators by the normal ...
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Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?
For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes:
$$
\forall f,g,h\in G:hg(f)=h(g(f))
$$
Now suppose there is additional axiom, or constraint if you prefer, ...
- 505
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Translating first order statements about symmetric groups into the language of numbers and back
A question I was asked recently lead me to the following question. For every closed first order formula $\theta$ in the group signature consider the set $N_\theta$ of natural numbers $n$ such that the ...
user6976
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A question on UCS p-groups
A $p$-group $G$ is called a ${\it UCS}$ $p$-group if $G$ has precisely three characteristic subgroups, namely $1$, $\Phi(G)$ and $G$.
Let $G$ be a finite UCS $p$-group of order $p^{2n}$ such that $\...
- 479
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Does a non-simple perfect group always have a maximal subgroup whose derived subgroup has nontrivial core?
Let $G \neq 1$ be a finite perfect group which is not simple.
Is it true that $G$ necessarily has a maximal subgroup whose derived subgroup
has nontrivial core in $G$?
Remark 1: This holds for all ...
- 347
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(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples
Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...
- 3,612
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An isomorphism between different Ext's coming from group cohomology
Let $G$ be an abelian group and $M$ a $G$-module with trivial action. It is well-known that $H^2(G,M)$ classifies extensions of $G$ by $M$, which is $\mathrm{Ext}^1_{Ab}(G,M)$.
On the other hand $H^2$...
- 15.4k
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Smallest subgroup of unitary group, containing diagonal matrices and a fixed unitary matrix is the whole group
Suppose that $U_n(\mathbb{C})$ is the group of unitary matrices of dimension $n$ over complex numbers. Fix a unitary matrix $A \in U_n(\mathbb{C})$ and consider the smallest closed subgroup $K \...
- 85
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Totally right preorderable groups
Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group?
More precisely:
totally right-preorderable: has a non-trivial total right-...
- 2,555
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Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$?
Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set
$$\{(x_1,\dotsc,x_{...
- 2,118
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Property of IA automorphisms of free groups
For $n \in\mathbb{N}$ define:
$X_n=\{x_1,\ldots,x_n\}$,
$F(X_n)$ the free group on $X_n$,
$\varphi:F(X_n)\to F(X_{n-1})$ an epimorphism defined by $x_i\stackrel{\varphi}{\mapsto} x_i$ for $1\le i\le ...
- 53
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How do "Kummer closures" of fields look?
Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...
- 54.6k
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Groups of order $p(p^2+1)/2$
It seems that when $p>3$ is a prime, then each group of order $p(p^2+1)/2$ is abelian as I checked by Gap for small $p$. Is it true for each $p$?
Thanks for your answers
- 1,168
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Generating infinite index subgroups of a free group
Let $F$ be nonabelian finitely generated free group, let $H \leq F$ be a finitely generated subgroup of infinite index and let $x,y \in F \setminus H$. Must there be some $a \in F$ such that $[F : \...
- 11.3k
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The category of subfactors extending the category of groups?
This post was inspired by this answer of Dave Penneys.
In the category of (irreducible hyperfinite II$_1$) subfactors, the morphisms of $(N \subset M)$ to $(N' \subset M')$ are usually defined as ...
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Convex sets on the discrete Heisenberg group
I'm interested in whether the finitely-generated discrete Heisenberg group admits a notion of "convex set". Below a formalization of what I need from the convex sets, in particular they should all be ...
- 6,652
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What is a "cusp" ("кусок") in relation to Guba's embedding theorem?
I'm confused by the definition of a "cusp" as found in
V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).
In the words of Mark ...
- 10.5k
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Do all right orderable groups have the Haagerup property?
Do all right orderable groups have the Haagerup property?
Recall that a group is right orderable if there exists a total order $\leq$ on it such that $a\leq b\Rightarrow ac\leq bc$. This property is ...
- 179
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Finite Unipotent Groups: References
It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)
The number of ...
- 1,169
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Non-associative commutative "group"
When dealing with some hash functions that I was trying to speed up, I toyed with a binary operation with the goal to "approximate" the addition on $\{0,1\}^*$ when seen as binary ...
- 48.9k
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Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
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Number of double cosets of a Young subgroup
Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$.
Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
- 43
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Minimal number of generators of satellite knot groups
In light of Knot groups with big number of generators, I was wondering...
Question 1 What is the minimal number of generators of the fundamental group of a satellite knot?
Another more specific ...
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