## New answers tagged gr.group-theory

0
votes

Accepted

### Reflections on subspaces of $\text{codim} > 1$

I found too late that this question is probably better off at MSE. However, for the sake of completeness and since I didn't find the formula in the literature, I will put it here:
$$s_{x_1, \dots, x_n}...

1
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### $G$-module structure of the relation module for a presentation of a finite group $G$

I think we may be able to extend the result to fields if the characteristic of the field $F$ does not divide the order of the group $G$.
Proposed Theorem: Let $F_n$ be a free group of rank $n \geq 2$ ...

4
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### The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group

It turns out that my original question does indeed have a positive answer. In fact, one can show that if $G$ has an irreducible character of degree $\geq 3$ then ${\rm AD}(G) \geq 2+ |G'|^{-1}$.
The ...

Community wiki

6
votes

### $G$-module structure of the relation module for a presentation of a finite group $G$

The question has been answered over fields of characteristic $0$ but not over $\mathbb Z$. as originally asked. It turns out that the statement is never true for $G$ noncyclic. It is proved in Lemma ...

11
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Accepted

### $G$-module structure of the relation module for a presentation of a finite group $G$

Your memory is correct, at least if you replace $\mathbb{Z}$ with a field $k$ of characteristic $0$. This is a theorem of Gaschütz. See
W. Gaschütz,
Über modulare Darstellungen endlicher Gruppen, ...

3
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### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

This is not a complete answer, but shows how this can be calculated from the character table (at least in this case) without using the usual Frobenius-Schur formula $\nu(\chi) = \frac{1}{|G|} \left( \...

9
votes

### What are the Schur indices of irreducible representations of $\operatorname{SL}(2,p)$?

If $p \equiv 1$ mod $4$ then all faithful irreducibles have Schur index two, and Schur indicator $-1$. If $p \equiv 3$ mod $4$, then most do, but there are two irreducibles of dimension $(p+1)/2$ with ...

2
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Accepted

### If $F$ is a prosoluble subgroup of a free profinite product $\amalg G_i$ and $F \cap G_i^g$ is pro-$p$, is also $F$ pro-$p$?

Let $I = \{1,2\}$ and let $G_1 = G_2$ be groups of order $p=2$. Their free product is $G = \langle \delta, \varepsilon |\, \delta^2= \varepsilon^2=1\rangle$, which is $\langle \tau, \varepsilon |\, \...

5
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### Relations between relations in the positive braid monoid

I found a published reference! This is the main result of:
Fukushi, Takeo, On a braid monoid analogue of a theorem of Tits., SUT J. Math. 47, No. 1, 45-53 (2011). ZBL1235.20036.
I'll keep my write up ...

7
votes

### Is any representation of a finite group defined over the algebraic integers?

I just stumbled across this ancient question, and I want to point out my notes here that prove the result in question. The proof is basically the same as moonface's accepted answer, but with two ...

9
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### Regular orbits for automorphisms of finite simple groups

As pointed out by Michael Giudici the answer is given by a result of Horoševskiĭ. Here is a proof following the paper by Horoševskiĭ.
Lemma: Let $\phi$ be an automorphism of $G$ with $|\phi|$ ...

10
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### Regular orbits for automorphisms of finite simple groups

By a result of Horoševskiĭ you can never find such an automorphism, that is all automorphisms of finite simple groups have a regular orbit.

3
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Accepted

### Fixed points free automorphisms of Teichmüller spaces

Yes. In a bit more detail: if the Teichmüller space has positive dimension then the given topological surface admits a pseudo-Anosov homeomorphism. (This is an exercise, but perhaps a non-trivial one,...

4
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### Regular orbits for automorphisms of finite simple groups

If I am reading your question correctly, then I think $A_{5}$ is an example where this fails. The automorphism group is isomorphic to $S_{5}$. The only elements of composite order in the automorphism ...

4
votes

Accepted

### Relation between Floyd and Gromov boundaries of hyperbolic groups

For any Floyd function $f$ (as in Karlsson's paper) not decaying exponentially too fast, the Gromov and Floyd compactifications indeed coincide. In fact, there is a more general result for relatively ...

4
votes

Accepted

### Are all "almost projective" groups free?

Yes.
As noted by YCor in the comments, both for the category of groups and the category of finitely generated grups if we take $G$ to be a free group (with the rank equal to the minimal size of a ...

Community wiki

9
votes

Accepted

### Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}$If I have chased through the literature correctly, I think the answer to your question is "yes". Specifically:
Dyer–Formanek–...

9
votes

Accepted

### What are double groups mathematically?

As far as I can tell, a double group is a double cover of a group. Specifically, if $G \subset \operatorname{SO}(n)$ is a group acting by rotations of $n$-dimensional space, its double group is the ...

9
votes

Accepted

### Shortest almost trivial element of free group

Repeating from the comments section:
This (natural and beautiful) question was previously asked and answered on this site. See Collapsible group words. It also appeared recently on math.se.
The ...

3
votes

Accepted

### An interior cone condition for Teichmuller spaces

Many (most?) points of the boundary do not satisfy the interior cone condition. See the paper Spirals in the boundary of slices of quasi-Fuchsian space by Goodman.

10
votes

Accepted

### Nonisomorphic central products on the same pair of groups?

The smallest example: $G = H = \mathbb{Z}/4 \times \mathbb{Z}/2$, generated by say $x$ of order 4 and $y$ of order $2$, and $A = B = \langle x^2, y \rangle \cong \mathbb{Z} / 2 \times \mathbb{Z} / 2$. ...

Community wiki

0
votes

### Is group theory useful in any way to optimization?

Do you count problems from shape analysis to be a relevant optimisation problem? Here you try to find for example the distance between shapes (=unparametrised curves) in euclidean space to each other. ...

Community wiki

12
votes

Accepted

### Distinct characters with the same character values, outer automorphisms and Galois conjugation

Take $G = S_3 \times S_4$ and consider the unique two-dimensional irreducible representation of $S_3$ and the unique two-dimensional irreducible representation of $S_4$. These have the same character ...

2
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### Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?

The question is answered by Will Brian arXiv, Feb. 6 2024.

12
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### Prove these are not surface groups

Here is an argument inspired by but not actually using one-relator group theory. Let's write $x_0$ instead of $a_1$ and $t$ instead of $b_1$ and then Tietze transform the presentation to
$$ \Gamma_{g, ...

7
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### Prove these are not surface groups

One can do this using group cohomology, specifically the symplectic form $H^1(G,\mathbb Z)\times H^1(G, \mathbb Z) \to H^2(G,\mathbb Z)$ arising from the cup product. I guess this is closely related ...

5
votes

Accepted

### Distribution of 2-groups

The following is an empirical argument to show that the total number of (isomorphism classes of) groups of order less than $2^m$ is dwarfed by the number of order exactly $2^m$.
The number of groups ...

7
votes

Accepted

### Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup

This is answered in the paper of Harvey Blau, "On Trivial Intersection of Cyclic Sylow Subgroups" Proc AMS 1985. Whenever a Sylow $p$-subgroup of a finite simple group is cyclic, it is T.I. ...

3
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### Compact subgroups of a linear group over non-Archimedean local field

The group $G={\rm GL}(n,F)$ acts transitively on $\mathcal O$-lattices of $F^n$. The stabilizer of the standard lattice $L_o = {\mathcal O}^n$ is $K={\rm GL}({\mathcal O})$. It is an open compact ...

4
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### Computing homology groups with GAP

Graham Ellis would be able to better comment on the correctness of his code for $SL(5,\mathbb Z)$, as he appears to be the author of the HAP package in GAP.
But his code executes quickly and claims to ...

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