# Group action on a set?

1. Feb 16, 2009

### tgt

1. The problem statement, all variables and given/known data
Up to isomorphism, how many transitive Z6={0,1,2,3,4,5} sets X are there? In other words Z6 acting on X. How many X, up to isomorphism are there?

2. Relevant equations
A key theorem is Let X be a G-set and let x in X. Then |Gx|=(G:G_{x}).

3. The attempt at a solution
I found 1 set with 1 element. 1 set with 2 elements. 2 sets with 3 elements. 5*5! sets with 6 elements.

However the answer only had 1 set with 1 element. 1 set with 2 elements. 1 set with 3 elements.

Surely there should be at least a set with 6 elements?

2. Feb 16, 2009

### Hurkyl

Staff Emeritus
There indeed should be a transitive G-set with 6 elements. Unless, of course, you missed a condition of the problem....

As for your counts, remember that you were asked to count isomorphism classes....

3. Feb 16, 2009

### tgt

Didn't leave out any of the problem. It is on p196 q17 of Fraleigh's book.

Yes, I see up to isomorphism. Take the set with 3 elements. I am able to produce to different multiplication tables with the same three elements in the set. That must mean two non isomorphic sets.

4. Feb 16, 2009

### Hurkyl

Staff Emeritus
I'm not convinced. What are your two examples?

5. Feb 17, 2009

### tgt

Attached is the two tables. Once the fixed elements and the multiples of 1 line is determined, the rest of the table is self explanatory by the axiom that a(bx)=(ab)x.

#### Attached Files:

• ###### two tables.doc
File size:
32 KB
Views:
59
6. Feb 19, 2009

### tgt

What do people think?

7. Feb 23, 2009

### tgt

Hurkyl, have you disappeared?

8. Feb 23, 2009

### Hurkyl

Staff Emeritus
I didn't want to download a .doc file, so I was hoping someone else who wasn't bothered would chime in.

9. Feb 28, 2009

### tgt

Anyway, here it is without the .doc file

* a b c
0 a b c
1 b c a
2
3 a b c
4
5

* a b c
0 a b c
1 c a b
2
3 a b c
4
5

The above are the two different tables hence two different isomorphisms.

10. Mar 1, 2009

### Hurkyl

Staff Emeritus
But those are isomorphic Z6-sets. One isomorphism is

a |--> a
b |--> c
c |--> b