Consider group G acting on a set X
Give examples of:
a)The action that is transitive and faithful
My Answer: Group G under addition acting on a set of integers Z
b)The action that is transitive but is NOT faithful
My Answer: Group G of 60 degree rotations acting on a set of Vertices of Dihedral group $D_3$ since all 3 rotations fix everything.
c)The action that is NOT transitive but IS faithful
Can I simply say it's just a group of symmetries of a single point? Since it only has 1 element it can't be transitive, right?
d) The action that is NOT transitive and NOT faithful
Something that is non abelian? I don't really know.
e) The action with 2 orbits
A line with vertices 1 and 2. - The group of symmetries acting on a set of vertices.
f) The action with 3 orbits
I have a triangle in my mind, but rotational symmetries of a triangle are stabilizers aren't they? Same goes for part (e).
a)The action that is transitive and faithful
Your Answer: Group G under addition acting on a set of integers Z
It's not an answer because you did not say how it is acting (see also Leon Aragones'comment). By the way, $\mathbb{Z}$ is always acting in a natural way on $G$ when $G$ is abelian (but the action might not be transitive).
My Hint (trivial) : The trivial group $G$ acts on $\{1\}$ trivially, this should do the job why ?
My Hint (a little less trivial) : Take $G$ a group acting on itself in some particular way...
b)The action that is transitive but is NOT faithful
Your Answer: Group G of 60 degree rotations acting on a set of Vertices of Dihedral group $D_3$ since all 3 rotations fix everything.
"set of Vertices of Dihedral group $D_3$" do you mean the set of vertices of the equilateral triangle $T$ for which $D_3=Isom(T)$? In that case there are 2 rotations which does not fix everything.
My Hint (trivial) : You should go more simple, take $X:=\{1\}$ (i.e. a set with one element) then a group $G$ always acts on $X$ (the action is unique why?). Under which conditions on $G$ is the action transitive? faithful?
My Hint (a little less trivial) : Take $G$ be a group with at least one proper subgroup $H$ which is not trivial then $G$ acts naturally on $G/H$ (left $H$-cosets). This should do the job.
c)The action that is NOT transitive but IS faithful
Can I simply say it's just a group of symmetries of a single point? Since it only has 1 element it can't be transitive, right?
It seems about right but you do need to give the whole set up.
My Hint (trivial) : Take $G$ to be the trivial group and $X$ be any set. Then there is only one action of $G$ on $X$. Under which condition on $X$ is the action transitive? faithful?
My Hint (a little less trivial) : Take $G:=GL_2(\mathbb{R})$, I claim that there is a natural action on $\mathbb{R}^2$, this one do the job.
d) The action that is NOT transitive and NOT faithful
Something that is non abelian? I don't really know.
Go simple.
My Hint (trivial) : take $G$ a group and $X$ a set. Define the action of $G$ on $X$ by $g.x:=x$. Under which conditions on $G$ and $X$ is the action transitive? faithful?
My Hint (a little less trivial) : Take $G$ be a group with at least one proper subgroup $H$ which is not trivial then $G$ acts naturally on $G/H$ (left $H$-cosets). Then $G$ also acts diagonally on $G/H\times G/H$. This should do the job.
e) The action with 2 orbits
A line with vertices 1 and 2. - The group of symmetries acting on a set of vertices.
In that case the action has only one orbit, hasn't it?
My Hint (trivial) : Trivial group and a set with two elements.
My Hint (a little less trivial) : Think about a square and an axial symmetry.
f) The action with 3 orbits
I have a triangle in my mind, but rotational symmetries of a triangle are stabilizers aren't they? Same goes for part (e).
My Hint (trivial) : Trivial group and a set with three elements.
My Hint (a little less trivial): Think about an hexagon and an axial symmetry.