# Math Help - The group S6

1. ## The group S6

I have an example of such a group,
But I want to know
what another group is isomorphous? (otherwise known group which does not belong to S6)

2. Originally Posted by rtrt1
I have an example of such a group,
But I want to know
what another group is isomorphous? (otherwise known group which does not belong to S6)
Sorry, what are you asking? Are you wanting to know what the Sylow-3 subgroups of $S_6$ are isomorphic to?

3. ## yes

!

4. Originally Posted by rtrt1
!
Okay, well what is the order of a Sylow-3 subgroup of $S_6$? How many groups are there of this order? (HINT: there are only two. Why?)

You now just have to work out which of these two groups a Sylow-3 group is isomorphic to. This reduces to proving whether or not there exists an element of order 9 in your group (why?).

So, does there exist an element of order 9 in $S_6$?

EDIT: Alternatively, you can go back to the previous time you posted this question, here, just over two weeks ago, and do what I suggested then. That is, look at the group generated by $(123)$ and $(456)$. What is its order (from it's order you can tell if it is a Sylow-3 subgroup)? What group is it isomorphic to? (It's not cyclic, but it is abelian).

5. ## so

I understand that this sub group is not isomorphous to Z3XZ3
So which group is isomorphous?

6. Originally Posted by rtrt1
I understand that this sub group is not isomorphous to Z3XZ3
So which group is isomorphous?
Why is it not isomorphic to $\mathbb{Z}_3 \times \mathbb{Z}_3$?

7. ## Because

Because it is not cyclic ( but Z3 is Cyclic )

8. Originally Posted by rtrt1
Because it is not cyclic ( but Z3 is Cyclic )
Yes, $\mathbb{Z}_3$ is cyclic, but $\mathbb{Z}_3 \times \mathbb{Z}_3$ is not ( $V_4=K_4$, the Klein 4-group, is not cyclic, and is isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_3$).

9. ## so

So that the group I was looking for ...

10. Originally Posted by rtrt1
So that the group I was looking for ...
Yup.