New answers tagged

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}...
Bipolar Minds's user avatar
1 vote

$G$-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 $n \geq 2$ ...
Damien's user avatar
  • 146
4 votes

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 ...
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 ...
Benjamin Steinberg's user avatar
11 votes
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, ...
Andy Putman's user avatar
  • 43.2k
3 votes

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( \...
Geoff Robinson's user avatar
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 ...
Dave Benson's user avatar
  • 11.2k
2 votes
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 |\, \...
Dan Haran's user avatar
5 votes

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 ...
David E Speyer's user avatar
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 ...
Andy Putman's user avatar
  • 43.2k
9 votes

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 $|\phi|$ ...
testaccount's user avatar
10 votes

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.
Michael Giudici's user avatar
3 votes
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,...
Sam Nead's user avatar
  • 25.3k
4 votes

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 ...
Geoff Robinson's user avatar
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 ...
M. Dus's user avatar
  • 1,900
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 ...
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–...
HJRW's user avatar
  • 23.8k
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 ...
Theo Johnson-Freyd's user avatar
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 ...
Sean Eberhard's user avatar
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.
Sam Nead's user avatar
  • 25.3k
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 $A = B = \langle x^2, y \rangle \cong \mathbb{Z} / 2 \times \mathbb{Z} / 2$. ...
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. ...
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 ...
Will Sawin's user avatar
  • 133k
2 votes

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.
Mohammad Golshani's user avatar
12 votes

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, ...
Giles Gardam's user avatar
  • 2,776
7 votes

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 ...
Will Sawin's user avatar
  • 133k
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 ...
Dave Benson's user avatar
  • 11.2k
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. ...
Dave Benson's user avatar
  • 11.2k
3 votes

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 ...
Paul Broussous's user avatar
4 votes

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 ...
Ryan Budney's user avatar
  • 42.8k

Top 50 recent answers are included