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
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

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
9 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
8 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
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
  • 43.1k
5 votes

Relations between relations in the positive braid monoid

My conjectured list generates $\pi_1$. This proof is basically the one in Lemma 3 and Proposition 4 of Tits' paper that Sam Hopkins referenced in the comments, plus some facts from Garside theory. ...
David E Speyer's user avatar
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

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 ...
3 votes

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
  • 1,875
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
2 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
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

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