# G must have an element with no fixed point when there is only one orbit

• wheezyg
In summary, the conversation discusses a problem involving a finite group operating on a set with only one orbit. The relevant theorem states that the order of the orbit is equal to the index of the stabilizer group in the group. The individual is seeking feedback on their attempt at solving the problem and questions the validity of their argument. They also mention using Lang's Algebra for guidance.
wheezyg
I am studying for a modterm on Monday and asking for help on the homework questions I got WRONG on my problem sets (so I can hopefully improve my understanding and see my mistake). This is my reworked version of the incorrect HW problem and I would like to know if I am on the right track...

Problem
Let G be a finite group operating on a set S ( |S| >=2 ). Suppose there exists only one orbit. Prove there exists an x \in G which has no fixed point (ie xs \neq s for all s \in S)

Relevant theorem
Let G be a group operating on a set S and s \in S . Then the order of the orbit Gs is equal to the index of G_s (stabalizer group or isotropy group in Lang) in G

My attempt
Suppose every point in G has a fixed point. Then for every x \in G, xs=s \in S. From this (G:G_s) =1. **Since there is only one orbit, (G:G_s)=|S| which implies |S|=1, a contradiction.

I have a feeling that my argument falls apart at **. Any guidance to the flaws inmky logic and/or understanding would be useful. I am working from Lang's Algebra (graduate level).

I don't think the stabilizer group will be too useful here.
G_s = g \in G such that gs=s.
This statement of yours is wrong:
"suppose every point in G has a fixed point. Then for every g \in G, gs=s \in S."

you are getting your quantifiers mixed up. it should be

"suppose every point in G has a fixed point. Then for every g \in G, there exists s\in S such that gs=s \in S."

## 1. What does it mean for "G to have an element with no fixed point"?

Having an element with no fixed point means that there is an element in the group G that does not map to itself under any element of the group. In other words, there is no element in the group that when applied to this particular element, will result in the same element.

## 2. Why is it important for a group to have an element with no fixed point when there is only one orbit?

It is important because it allows us to distinguish between different orbits in the group. If there is only one orbit, then all elements in the group will map to each other under the group operation. However, if there is an element with no fixed point, it will not map to any other element in the group, thus creating a distinct orbit.

## 3. How does having an element with no fixed point affect the structure of the group?

Having an element with no fixed point can affect the structure of the group by creating distinct orbits, as mentioned before. This can also affect the group's properties and subgroups, as elements with no fixed point may not behave the same way as other elements in the group under certain operations.

## 4. Is it possible for a group to have only one orbit without having an element with no fixed point?

Yes, it is possible for a group to have only one orbit without having an element with no fixed point. This means that all elements in the group map to themselves under the group operation. In this case, the group is said to be "transitive".

## 5. How does the presence or absence of an element with no fixed point affect the group's symmetry?

The presence or absence of an element with no fixed point can affect the group's symmetry by creating distinct orbits and subgroups within the group. It can also affect the group's symmetry by changing the way certain elements behave under group operations. Additionally, the presence of an element with no fixed point can be used to classify groups into different symmetry types.

Replies
1
Views
1K
Replies
1
Views
858
Replies
1
Views
903
Replies
2
Views
971
Replies
3
Views
1K
Replies
3
Views
972
Replies
5
Views
1K
Replies
4
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K