Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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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 ...
Nick L's user avatar
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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,...
Bipolar Minds's user avatar
9 votes
1 answer
869 views

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 ...
Nguyen lan Lee's user avatar
4 votes
1 answer
313 views

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 ...
Neil Hoffman's user avatar
  • 5,221
1 vote
0 answers
585 views

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 ...
FrankMiller's user avatar
3 votes
2 answers
169 views

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 ...
R Maharaj's user avatar
  • 366
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 ...
Andy Putman's user avatar
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5 votes
0 answers
431 views

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 ...
user106317's user avatar
6 votes
1 answer
283 views

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 $T$. ...
Dirk's user avatar
  • 809
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 ...
Pablo's user avatar
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3 votes
0 answers
90 views

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 ...
Shahrooz's user avatar
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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 ...
Bedovlat's user avatar
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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 $\mathrm{Borel}(M)$ ...
Bedovlat's user avatar
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10 votes
3 answers
475 views

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 $W$...
THC's user avatar
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6 votes
0 answers
219 views

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 $k$ ...
user avatar
2 votes
2 answers
655 views

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 ...
Noam Kolodner's user avatar
-2 votes
1 answer
505 views

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( \...
john mangual's user avatar
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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, ...
user237522's user avatar
  • 2,783
2 votes
2 answers
936 views

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}, \...
john mangual's user avatar
  • 22.6k
1 vote
0 answers
191 views

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) ...
wonderich's user avatar
  • 10.3k
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 ...
Taras Banakh's user avatar
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5 votes
1 answer
328 views

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 ...
Arrow's user avatar
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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 ...
S. Razamat's user avatar
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 ...
Justin Benfield's user avatar
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 ...
Watson Ladd's user avatar
  • 2,419
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\, ...
john mangual's user avatar
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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 ...
R Maharaj's user avatar
  • 366
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 $A\cap B=Z(G)$, ...
Maryam's user avatar
  • 99
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 ...
Alireza Behtash's user avatar
6 votes
0 answers
454 views

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 ...
the_fox's user avatar
  • 347
14 votes
1 answer
676 views

$\mathbb{Z}$-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 $\...
user108921's user avatar
3 votes
1 answer
184 views

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 ...
Struggler's user avatar
  • 153
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 $\...
T. Amdeberhan's user avatar
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 \...
Pablo's user avatar
  • 11.2k
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. ...
letta's user avatar
  • 21
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 $\hat{G}$ ...
Pablo's user avatar
  • 11.2k
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 \...
Martin Brandenburg's user avatar
5 votes
1 answer
255 views

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 ...
Vladimir's user avatar
  • 1,210
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 ...
user43198's user avatar
  • 1,949
15 votes
1 answer
775 views

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 ...
Ali Taghavi's user avatar
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]...
T. Amdeberhan's user avatar
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 ...
user avatar
9 votes
1 answer
286 views

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. ...
Vipul Naik's user avatar
  • 7,230
5 votes
0 answers
163 views

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 $X$ ...
Mikhail Borovoi's user avatar
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 $\...
fritz's user avatar
  • 183
11 votes
1 answer
2k views

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 ...
pre-kidney's user avatar
  • 1,289
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....
M L's user avatar
  • 381
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 ...
Igor Rivin's user avatar
  • 95.5k
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 ...
Danny Neftin's user avatar
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 $\xi$ ...
David Cohen's user avatar

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