Group Action on a Set: Counting Transitive Z6 Sets Up to Isomorphism

[tgt]

Homework Statement

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?

Homework Equations

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

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?

 

[Hurkyl]

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...

 

[tgt]

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...

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.

 

[Hurkyl]

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

 

[tgt]

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

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.

 

[tgt]

What do people think?

 

[tgt]

Hurkyl, have you disappeared?

 

[Hurkyl]

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

 

[tgt]

Strange reason? Are you worried about your memory?

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.

 

[Hurkyl]

But those are isomorphic Z6-sets. One isomorphism is

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