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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).

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  • $\begingroup$ "My Answer: Group G under addition acting on a set of integers Z": you haven't specified how it is acting. $\endgroup$ – user207868 Apr 19 '15 at 18:35
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    $\begingroup$ I am afraid that you won't get many marks for any of those answers! $\endgroup$ – Derek Holt Apr 19 '15 at 18:41
  • $\begingroup$ Under addition (0 is the identity) and G is a set of all positive numbers $\endgroup$ – GRS Apr 19 '15 at 18:53
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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.

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  • $\begingroup$ Thank you very much, For (e) I imagine any $D_n$ let's say $D_3$. Then If I label the faces as Top, Bottom - the swap has 2 orbits.For (f) I imagined an object that looks like a rectangle but with triangular sides. For (a) could a symmetry of any $D_n$. (c) is a symmetry of a graph that only has 1 symmetry - the identity (acting on vertices). I'm still not clear about non transitive actions and what they mean though. $\endgroup$ – GRS Apr 25 '15 at 15:17

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