Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,090 questions
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Coxeter subgroups of Coxeter groups
Is there an algorithm to determine all the Coxeter subgroups of a given Coxeter group? If we only want the Coxeter subgroups of finite index does that make the question easier? If we only want a ...
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Does Aut(G) → Out(G) always split for a compact, connected Lie group G?
The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
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3
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Membership to double cosets in free groups
Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...
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Are countable FC-groups maximally almost periodic?
An FC-group is a group in which every element has a finite conjugacy class. A group G is said to be maximally almost periodic if there is an injective homomorphism from G into a compact Hausdorff ...
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Lower central series vs torsion-free lower central series
$\newcommand\tf{\text{tf}}\newcommand\tor{\text{tor}}$Let $G$ be a finitely generated group. Let $\gamma_k(G)$ denote the $k$th term in the lower central series for $G$, so $\gamma_1(G) = G$ and $\...
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Proofs of a character identity?
Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C_1,...,C_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum_{(c_1,...,c_k) \in C_1 \...
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Conjugcy classes in GL(F_2)? GL(F_q)
How to deduce a formula (see below) for number of conjugacy classes in $\operatorname{GL}_n(F_2)$? (More generally F_q) ?
Is there some description of conjugacy classes or we just know how many of ...
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What is the maximal possible rank of a subgroup of a special linear group mod a prime?
Let $p$ be a prime number, and let $\mathbb{F}_p$ be the unique field of cardinality $p$.
What is $\max \{d(H) : H \leq \mathrm{SL}_3(\mathbb{F}_p)\}$?
Here we denote by $d(G)$ the smallest ...
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The Higman group II
This question is related to the question The Higman group (with a nice answer by M. Sapir). So for background, please,
see the above cited question.
The Higman group has an automorphism $h(a_j)=a_{j+...
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Malcev Lie algebra and associated graded Lie algebra
Suppose $L$ is a nilpotent finite-dimensional Lie algebra over $\mathbb{Q}$ of class $c$. We can define an associated graded Lie algebra to $L$ that, as a vector space, is:
$$\bigoplus_{i=1}^c \...
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Are the distributive permutation groups linearly primitive?
An action of a group $G$ on a set $X \neq \emptyset$ is called transitive if $\forall x,y \in X$, $\exists g \in G$ such that $g.x = y$.
It is called primitive if it is transitive and preserves no non-...
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Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
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recognition of symmetric groups in GAP
In GAP (https://www.gap-system.org), there is a function IsSymmetricGroup, which tells you whether a subgroup of $S_n$ generated by given permutations is all of the $S_n$. It looks like it takes ...
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$G^F$ conjugacy class of $F$-stable maximal tori, in an algebraic group $G$ defined over $\mathbb{F}_{q}$
Let $G$ be an affine algebraic group over $k=\bar{\mathbb{F}_{p}}$. Let $q$ be a power of $p$, and assume that $G$ is defined over $\mathbb{F}_q$. Let $\mathcal{T}$, be the collection of all maximal ...
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Reversible varieties
We say that a variety $V$ is reversible if for each $n>0$ and $n$-ary fundamental operation $f$, there is some $m\geq n$ and $r$ along with terms
$T_{2},\dotsc,T_{r},S_{1},\dotsc,S_{m}$ such that $...
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Is the category of group objects in an $(\infty,1)$-topos reflective as a subcategory of the groupoid objects?
Let $C$ be an $(\infty,1)$-topos. The $(\infty,1)$-category of group objects in $C$ is a full sub-$(\infty,1)$-category of groupoid objects in $C$:
$${\mathsf{Grp}}(C) \hookrightarrow {\mathsf{Grpd}}(...
user2529
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Aspherical manifold with superperfect fundamental group and non-trivial center?
I am interested in knowing if there is a closed, (smooth) aspherical manifold $M$ (hyperbolic would be best) with superperfect fundamental group (that is to say, with $H_1(\pi_1(M);\mathbb{Z}) = H_2(\...
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Automorphism group of the special unitary group $SU(N)$
Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...
- 2,147
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Rotation numbers for amenable group actions on the circle
Given an orientation-preserving homeomorphism $f: S^1 \to S^1$, one can define its rotation number $\rho(f) \in \mathbb{R}/\mathbb{Z}$, as $\rho(f) = (\lim_{n \to \infty} \tilde{f}^n(0)/n) + \mathbb{Z}...
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Subgroup property stronger than being characteristic
In what follows, all groups are assumed to be finite.
Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a weakly closed subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ ...
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Consequences of the Inverse Galois Problem
Are there any papers written about the consequences of the Inverse Galois Problem in case it is proved to be true or false?
We know a lot of things that would be true if the Riemann Hypothesis holds. ...
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Transitive actions of $Aut(F_2)$ on surjections from $F_2\twoheadrightarrow G$
Here $F_2$ is the free group on two generators $x,y$. I'm interested in examples of finite groups $G$ such that $Aut(F_2)$ acts transitively on the set of surjections $F_2\rightarrow G$. (In ...
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Two-relator products of cyclic groups
In "A proof of the Scott–Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...
- 1,000
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mapping space between classifying spaces
I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces ...
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Homomorphisms from higher rank lattices with infinite center to $\mathbb{Z}$
Suppose that $\Gamma$ is an irreducible lattice in a semi-simple real Lie group $G$ of higher rank (with infinite center!), is every homomorphism $\Gamma \to \mathbb{Z}$ trivial?
The case where $G$ ...
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Is there an algebraically normal function from $\mathbb{Z}^{n}$ to $\{ 0 , 1\}$?
Definition: Let $h$ be a polynomial in $n$ variables, then :
$\gamma(h,r,R):=\{ v \in \mathbb{Z}^{n} : \vert h(v) \vert \leq r, \Vert v \Vert < R \}$
Let $\omega : \mathbb{Z}^{n} \to \{ 0 , 1\}$...
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Simplicity of infinite groups
Sorry about asking so many questions, but I am a bit further on in my classification, and I am up to the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10}, ([a,b]^4b)^7 \rangle$. It has no ...
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Automorphism group of a special commuting graph
Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets ...
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Which groups contain a comb?
The comb is the undirected simple graph with nodes
$\mathbb{N} \times \mathbb{N}$
where $\mathbb{N} \ni 0$ and edges
$$ \{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{N}, n \in \mathbb{N}...
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Centralizer of longest element in a finite irreducible Weyl group: related to folding of ADE graphs?
Say $(W,S)$ is a finite Coxeter group, such as a Weyl group (which satisfies an additional crystallographic condition). Assume also that $W$ is irreducible. Then it has a longest element $w_o$ ...
- 52.9k
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The kernel of all invariant means
Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in ...
- 4,432
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Simple groups and irreducible characters of degree 3
I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here.
The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}...
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When is a Schreier coset graph vertex transitive
When is a Schreier Coset graph on a group $G$ with subgroup $H$ and symmetric generating set $S$(without identity) vertex transitive?
It is well known that when $H$ is normal, the Schreier coset graph ...
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Are generalized symmetric groups maximal finite groups (in a certain sense)?
Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
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Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
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"Factorisation" in special linear groups over rings of integers
It is known that for any number field $F$ with infinitely many units (i.e. $F$ is not $\mathbb Q$ or an imaginary quadratic field) with ring of integers $O$ the special linear group $\mathrm{SL}_2(O)$ ...
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Pairs of Permutations up to Simultaneous Conjugation
The conjugacy classes of $S_n$ are the cycle types since if $\tau = (\dots)(\dots)\dots(\dots)$, the conjugation $\tau \mapsto \sigma \tau \sigma^{-1}$ permutes the labels in the cycles of $\tau$.
...
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Generic set that is a proper subgroup
For a group $G$ generated by a finite set $S$ we denote by $B_{G,S}(n)$ the ball of radius $n$, that is the set of all elements in $G$ which are expressible as products $x_1x_2\ldots x_n$ where $x_i\...
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Bernstein's presentation for the Hecke algebra
Any one know of any good references for reading about Bernstein's presentation of the Iwahori Hecke algebra? I need some notes which has an example or two. It would really help.
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A "direct" proof that hyperbolic groups are not amenable
I am looking for a proof that a finitely generated hyperbolic groups is non-amenable [unless it is virtually cyclic] which is as "metric/combinatoric" as possible.
Here are the two proofs I am aware ...
- 4,432
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How many finite simple groups of order $p+1$?
I'm looking at finite simple groups of order $p+1$ where $p$ is a prime number.
But they don't seem to fall into any classification - have these all been determined? Is the number of them even ...
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General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups
As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast
f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...
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Classification of $p$-groups of order $p^n$ with rank $n-1$
Hello,
i've been looking for a way to classify the non-trivial $p$-groups $G$ that live in an exact sequence of the form
$ 0 \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow G \rightarrow (\mathbb{Z}...
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Upper bound for size of subsets of a finite group that contains a sum-full set
Problem
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we
define $k(G)$ be ...
- 1,250
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von dyck groups and solvability
A von Dyck group is a group with presentation $< a,b | a^m=b^n=(ab)^p=1 >$ with m,n,p natural numbers. Is it known which of these groups are solvable and which are not? Is there a reference ...
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Amenable exponential growth
Dear forum members,
Does anyone have a clear example of an amenable group with exponential growth?
Is real that if G is virtually amenable (has an amenable subgroup of finite index) then it is ...
- 375
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Normal subgroups of braid groups
There is a lot of normal subgroups in braid groups (for example there is an action of $B_n$ on unitriangular bilinear forms on $R^n$ over arbitrary commutative ring $R$: $b_i\colon e_j\mapsto e_j$, $j\...
- 1,422
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Nilpotent group with ascending and descending central series different?
This may turn out to be a bit embarassing, but here it is. It is probably not true that the ascending and descending central series (*) of a nilpotent group have the same terms. (Or at least one of ...
- 1,570
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Cayley graphs and its subgraphs
I have two questions about Cayley graphs. Any answers will be appreciate.
1) Do we have any Cayley graph that has Petersen graph as its induced subgraph?
2) Suppose $Cay(G,S)$ be a Cayley graph that ...
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Can every nonempty set carry abelian group structure? [duplicate]
Possible Duplicate:
Does every non-empty set admit a group structure (in ZF)?
Let $X$ be an arbitrary nonempty set. Can you define a multiplication making it into an abelian group?
If $X$ is ...
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