Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
7,922
questions
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Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
1
vote
0
answers
98
views
Symmetric analogue of "alternating bihomomorphism is skew of 2-cocycle" theorem
Let $G$ be a finite abelian group. It is well-known that every alternating bihomomorphism $\Omega:G\times G \to \mathbb{C}^\times$ arises as the skew $\kappa/\kappa^T$ of a 2-cocycle $\kappa \in Z^2(G,...
9
votes
1
answer
869
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Kaplansky conjecture (consequences)
The Kaplansky conjecture says that: for any field $F$ and any torsion free group $G$, the group ring $F[G]$ does not have nontrivial idempotent elements.
Questions
Do we assume that $F$ has any ...
4
votes
1
answer
313
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For which groups is (non-)left orderability decidable?
Mainly, my question is in the title, but let me be more precise here.
Let $G$ be a finitely presented group with solvable word problem. If G is not left-orderable, is there an finite-time algorithm ...
1
vote
0
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585
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Inverse limits and first isomorphism theorem for compact topological groups
This question was originally asked on MathSE here.
I have a problem with Proposition (1.2.1) from J. Wilson's book 'Profinite Groups'
The proposition is the following:
Let $(G, \varphi_i : G \to ...
3
votes
2
answers
169
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Weak Pronormality of a finite group
Definition: A subgroup $H$ of $G$ is said to be pronormal in $G$ if for all $g\in G$, there exists $x \in \langle H, H^g \rangle$ such that $H^x =H^g$.
Definition: A subgroup $H$ of $G$ is said to be ...
11
votes
2
answers
930
views
Subtle point in definition of BNS invariant
Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling ...
5
votes
0
answers
431
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Subgroups and quotients of an abelian pro-finite group
It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$.
I'm wondering whether there is a counterpart for profinite groups.
For example is it true ...
6
votes
1
answer
283
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Reference request: Reduced reflection length in Coxeter groups
I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
4
votes
1
answer
413
views
Finite index subgroups of a RAAG
Let $G$ be the group given by the presentation
$$\langle x,y,z,w \ | \ xy = yx, yz = zy, zw = wz\rangle.$$
This is a right-angled Artin group (RAAG) whose graph is a path on $4$ vertices.
We can ...
3
votes
0
answers
90
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Visualization (even locally) of graphs with given infinite group
I want to give a lecture about Frucht theorem (and its generalization) which state that: for each finite group $G$, there is at least one finite graph $\Gamma$ such that $Aut(\Gamma)\cong G$. For each ...
12
votes
1
answer
559
views
Can a simple group be equivalent to a non-simple group?
Two abstract groups $G$ and $H$ are called equivalent, $G\sim H$, if each of them is isomorphic to a subgroup of another.
Question: Can a simple group $G$ be equivalent to a non-simple group $H$?
Of ...
11
votes
2
answers
549
views
Homeomorphisms vs Borel automorphisms
Let $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ be the groups of homeomorphic and Borel automorphisms of a space $M$, respectively.
Question: Are $\mathrm{Homeo}(M)$ and $\mathrm{Borel}(M)$ ...
10
votes
3
answers
475
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From Gassmann-Sunada triples to isospectral manifolds
A Gassmann-Sunada triple is a triple $(U,V,W)$ of groups, with $V, W$ subgroups of $U$, such that $U$ and $V$ meet every conjugacy class in $U$ in the same number of elements, and such that $V$ and $W$...
6
votes
0
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219
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Lower bound for order of matrix modulo $n$
For a positive integer $n$ and a square matrix $A$ with integral entries, let $\text{ord}(A, n)$ be the smallest positive integer $k$ such that $A^k \equiv \mathbf{1} \bmod n$, if such integer $k$ ...
2
votes
2
answers
655
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Subgroup of a free group that is characteristic but not totally characteristic
Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...
-2
votes
1
answer
505
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no classification of nilpotent lie groups
there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\...
7
votes
1
answer
457
views
The group of $k$-automorphisms of $k[x_1,\ldots,x_n,x_1^{-1}]$
Let $k$ be a field (of characteristic zero).
For $k[x_1,\dotsc,x_n]$ it is known that the affine and triangular automorphisms generate $G_n$, the group of automorphisms of $k[x_1,\dotsc,x_n]$,
see, ...
2
votes
2
answers
936
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are all simply connected nilpotent lie groups matrix groups over $\mathbb{R}$?
Mathworld just says that the lower central series terminates: $\mathfrak{g}_1= [ \mathfrak{g}, \mathfrak{g}]$,
$\mathfrak{g}_2= [ \mathfrak{g}, \mathfrak{g}_1]$ and $\mathfrak{g}_n= [ \mathfrak{g}, \...
1
vote
0
answers
191
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Non-existence of nontrivial finite group extension of any simply-connected Lie group
Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that
there does not exist any group $G$ (with no topology) ...
4
votes
1
answer
330
views
Is there a topologizable group admitting only Raikov-complete group topologies?
Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is ...
5
votes
1
answer
328
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Short proof a monoid is a group iff every splitting is right homogeneous
In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum
June 2014, the authors prove a characterization of groups among ...
5
votes
1
answer
428
views
Identify one group of linear transformations
Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Q})$ defined as follows:
Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Q}^9$, $G$ is generated by:
All ...
5
votes
2
answers
469
views
Wielandt automorphism tower theorem
I wanted to know if anyone can point me to an (ideally freely available) english translation of the proof of Wielandt's Automorphism Tower Theorem (1939).
The theorem states the following:
Given a ...
2
votes
0
answers
63
views
Determining subgroup of finite group of Lie type from characteristic polynomials
Suppose you have $G$ a finite group of Lie type (say $\mathrm{Sp}_4( \mathbb{F}_5)$ as a case I particularly care about, but there are others.) Your friend picks a subgroup $H$ and selects random ...
1
vote
2
answers
311
views
group theory behind the Kloosterman bound $| S(m,n;c) |< 2\, c^{3/4}$
I am trying to understand Kloosterman sums and their estimates (e.g. from [1], which does not prove)
$$ \Big| S(m,n;c) \Big| = \Big| \sum_{(x,c) = 1} e\big( \frac{mx + nx^{-1}}{c}\big) \Big| < 2\, ...
1
vote
1
answer
178
views
The order of the system normalizer in a finite solvable group
Definition: Let $G$ be a finite solvable group and $\Sigma \in \text{H}(G)$, the set of Hall systems of $G$. The normaliser of $\Sigma$ is defined as $$ N_G(\Sigma) = \{ g\in G \,|\, H=H^g \text{ for ...
2
votes
1
answer
195
views
p-group as a product of two abelian normal subgroups
Thanks for any comment or answer.
Let $G$ be a finite non-abelian $p$-group such that $G=AB$ where $A=C_G(a)$ and $B=C_G(b)$ are maximal abelian normal subgroups of $G$ such that $A\cap B=Z(G)$, ...
2
votes
0
answers
590
views
Volume of $SL(2,\mathbb{C})$ [closed]
So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf
I can write the Haar measure of $SL(2,\mathbb{C})$ as
$$d\mu = \sinh^2(r) dr dk dk'$$
where $r$ runs over nonnegative real numbers ...
6
votes
0
answers
454
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On the average number of subgroups per conjugacy class
At some point in the future, I hope to do some work on estimates for the number of conjugacy classes of subgroups of a finite group (by an estimate here, I mean an upper bound). Assuming, for the ...
14
votes
1
answer
676
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$\mathbb{Z}$-module structure of the subring generated by an algebraic number
Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\...
3
votes
1
answer
184
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Finite congruence-free semigroup without zero [closed]
I am reading the book " Fundamentals of semigroup theory" by John M. Howie $\textbf{(Section 3.7)}$.
I want to prove $\textbf{Theorem 3.7.2}$
If $S$ is a finite congruence free semigroup ...
2
votes
2
answers
242
views
$n$-distant permutations more than not
Let $\mathfrak{S}_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}_{2n}$ has an $n$-distant pair if there is some $j\in [2n-1]$ such that $\...
12
votes
2
answers
677
views
Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...
2
votes
0
answers
88
views
Example of action of an infinitely countable group that has important ergodic/statistical property?
I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. ...
8
votes
1
answer
598
views
Is there a residually finite non-elementary hyperbolic group whose profinite completion is boundedly generated?
Is there a residually finite hyperbolic group $G$ that is not virtually cyclic, such that there exists finitely many procyclic closed subgroups $C_1, \dots, C_n$ of the profinite completion $\hat{G}$ ...
8
votes
2
answers
478
views
Exact sequence of $n$th powers of abelian groups
Let $A,B,C$ be finitely generated abelian groups. Assume that there is an exact sequence $$0 \to C \to A^n \to B^n \to 0,$$where $A^n = A \oplus \dotsc \oplus A$ as usual. It is not assumed that $A^n \...
5
votes
1
answer
255
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Can an amenable group have a weak mixing unitary representation without almost invariant vectors?
Does there exist a finitely generated discrete amenable group $G$ that acts on a separable Hilbert space $\mathcal{H}$ by unitary transformations, and where (1) $\mathcal{H}$ has no finite dimensional ...
2
votes
0
answers
68
views
Centralizer/Normalizer of global sections of vector bundles on curves
Let $X$ be a smooth, projective curve of genus at least $2$ over $\mathbb{C}$ and $E$ be a vector bundle on $X$ of rank at least $2$. Given any point $x \in X$, denote by $S_x$ the image of the ...
15
votes
1
answer
775
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The completion of the space of finite groups
Edit: I revise the question based on the comment conversations
Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation.
We define ...
5
votes
2
answers
335
views
Determinant of the "quantum" version of the group $\mathbb{Z}_n$
Let $[0]_q:=0$ and $[n]_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.
Question. Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries
$$[i+j\bmod n]...
9
votes
1
answer
153
views
Inductive and reducible functions
The question was asked by a Computer Scientist and is closely related to parallel computing. But it is clearly of algebraic nature, so I decided to post it here.
Let $X$ be a set and $\bar X$ be the ...
9
votes
1
answer
286
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Example of 2-locally finite group that is not locally finite
Define a group to be 2-locally finite if, for any two elements, the subgroup generated by them is finite.
Define a group to be locally finite if the subgroup generated by any finite subset is finite.
...
5
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0
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163
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The group of automorphisms a pair $(G,X)$ where $X$ is a spherical homogeneous space of $G$
I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$.
First let $G$ be any algebraic group over $\mathbb C$, and let $X$ ...
7
votes
0
answers
352
views
Let $G$ be a group of finite cohomological dimension. $\mathbb Z$ normal in $G$. Is it true that $cd_{\mathbb Q}(G/\mathbb Z) < cd_{\mathbb Q} (G)$?
$cd_{\mathbb Q}$ is the cohomological dimension on rational coefficients. Notice that if we replace $\mathbb Q$ with $\mathbb Z$ in $cd_{\mathbb Q}$ the statement is certainly false (think about $\...
11
votes
1
answer
2k
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Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
2
votes
0
answers
85
views
Automorphisms of a free topological product
Let $G$, $G_1$, $G_2$ be Hausdorff topological groups. I am mainly interested in the case when those groups are profinite.
Let $G$ act continuously on $G_1$ and $G_2$ via continuous automorphisms, i.e....
13
votes
0
answers
354
views
Euler characteristic of *hyperbolic* orbifolds
This is a follow-up to this question. Which rational numbers arise as Euler characteristics of orbifold quotients of $\mathbb{H}^n?$ The answer is not even clear for $n=2.$ It is clear that the Euler ...
27
votes
0
answers
876
views
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 ...
3
votes
0
answers
103
views
Degree of a local cut point in the boundary of a hyperbolic group
Suppose $G$ is a one-ended word-hyperbolic group and $\xi$ is a (local) cut point of $\partial G$. Fix any visual metric on $\partial G$ and let $U(\epsilon,\xi)$ be the connected component of $\xi$ ...