5
votes
2answers
58 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
2
votes
3answers
45 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
5
votes
2answers
191 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
1
vote
1answer
83 views

The group of rigid motions of an icosahedron.

Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$. Can you help me to prove this? What I have done is shown that the order of the group of rigid motions of icosahedron is ...
1
vote
3answers
91 views

The small dihedral groups $D_1$ and $D_2$.

In M. Artin's book Algebra he wrote: But I think this visualisation $D_1$ and $D_2$ is inconsistent and confusing, because I guess $D_n$ could be associated with a subgroup of $S(\{1, \ldots, n\})$, ...
7
votes
1answer
155 views

On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation ...
4
votes
1answer
254 views

I don't understand symmetries of the Fano plane

Hello I got a picture of the Fano plane, but there are 5 points on every line. Why aren't there 3? And I cannot see what it's symmetry group is. I have been told it's $PSL_2(7)$ but that doesn't ...
16
votes
3answers
568 views

Is there a geometric realization of Quaternion group?

Is there a geometric realization of the Quaternion group: $$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$ I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
1
vote
0answers
27 views

Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
0
votes
0answers
28 views

Building invariant $S_n$ structures from two invariant $Z_n$ structures

Take two mathematical structures with a $Z_n$ symmetry (cyclic symmetry). Which are the different ways, in "gluing" these structures, to obtain a mathematical structure with a $S_n$ symmetry ...
6
votes
1answer
242 views

Solid whose full symmetry group corresponds to $A_4\times\mathbb Z_2$

So, as per one of my previous questions, I'm working through some problems in Armstrong's book Groups and Symmetry. The first two thirds of the question I've managed to grind through (after much ...
4
votes
2answers
368 views

Geometrical meaning of automorphisms of cyclic groups

I'm looking for a geometrical interpretation of the action of automorphisms of cyclic groups. I'll take one particular example to make it clear : I'm taking the cyclic group $\mathbb{Z}_{12}$, which ...