16
votes
2answers
871 views

Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials in the $A_5$ symmetries of the icosahedron (or dodecahedron)? Perhaps this is too vague a question. Q2. Are there ...
1
vote
1answer
246 views

Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal isotropics of $V\bigotimes \mathbb{C}$ ?

We say $L< (V\oplus V^{*})\bigotimes \mathbb{C}$ is isotropic when $< X,Y>=0$ for all $X,Y\in L$ Why $O(4n,\mathbb{C})$ (orthogonal group) acts transitively on the space of maximal ...
4
votes
3answers
261 views

lines through A_n reflection arrangement and permutations

(updated; apologies for way too much room left for interpretation in the original post) Let $\mathcal{A} =A_{n-1}$ be the $A_{n-1}$ arrangement in $\mathbb{R}^{n}$, i.e. the set of hyperplanes ...
9
votes
3answers
405 views

Recognising group actions on trees from the boundary

Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). ...
5
votes
2answers
195 views

Normal form(s) for the elements of hyperbolic triangle groups

I'm seeking a reference or a sketch for any sort of normal form that would enable rapid enumeration without redundancies of the elements of hyperbolic triangle groups and/or von Dyck groups.
0
votes
1answer
318 views

Action of Isometries on a Line in the Plane

I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...
8
votes
0answers
294 views

is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
2
votes
3answers
285 views

Local finiteness and coarse bounded geometry

I've just started learning these things and so probably my questions will be very easy. Please forgive me. A metric space $(X,d)$ is called locally finite if every bounded set is finite. A metric ...
5
votes
0answers
253 views

Erlangen program carried out explicitely?

I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations. What I mean by "carrying out Erlangen program" is picking a specific group (say ...
7
votes
0answers
357 views

Who conjectured that a transitive projective plane is Desarguesian?

The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved. ...
0
votes
1answer
206 views

Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$

Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to ...
5
votes
2answers
657 views

Geometric interpretation of $BN$-pairs

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres). ...
11
votes
7answers
2k views

Cheap, non-constructive, free group generating rotations for Banach-Tarski

Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group. For teaching purposes ...
7
votes
1answer
974 views

Burnside's Lemma and Geometry

I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action. I wonder if there is any ...
10
votes
2answers
1k views

Assistance with understanding parent/child relationships in Pythagorean Triples

Hello, I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained ...
7
votes
2answers
304 views

Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module?

Can any properties of a ring other than being a field be captured by the geometry of its 2-dimensional free module? Background: In his wonderful, wonderful book Geometric Algebra, Emil Artin ...
14
votes
3answers
836 views

Use of n-transitivity in finite group theory

Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups Sn or the alternating groups An+2, see e.g. page 226 this book by Isaacs ...