# Tagged Questions

**16**

votes

**2**answers

895 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

**1**answer

252 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

**3**answers

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

**3**answers

416 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

**2**answers

198 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

**1**answer

338 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

**0**answers

304 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

**3**answers

295 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

**0**answers

255 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

**0**answers

365 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

**1**answer

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

**2**answers

661 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

**7**answers

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

**1**answer

975 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

**2**answers

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

**2**answers

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

**3**answers

842 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 ...