Finding 2-Sylow & 3-Sylow of ##S_4##

  • Thread starter Lee33
  • Start date
To find the 2-Sylow subgroup of S_4, we note that it has order 8, and the only subgroups of order 8 in S_4 are isomorphic to the dihedral group of order 8.
  • #1
Lee33
160
0

Homework Statement



Find the three 2-Sylow subgroups of ##S_3## and find a 2-Sylow subgroup and a 3-Sylow subgroup of ##S_4.##

2. The attempt at a solution

I got ##|S_3| = 6 = 2\dot\ 3## and ##|S_4| = 24 = 2^3\dot\ 3.## So ##S_3## has a a Sylow 2 subgroup of order 2 and a Sylow 3 subgroup of order 3. I am asked to find the three 2-Sylow subgroups of ##S_3## so since the 2-Sylow of ##S_3## has order 2 is it just the permutations ##(1 2), (1 3), (2 3)? ##
 
Physics news on Phys.org
  • #2
Lee33 said:

Homework Statement



Find the three 2-Sylow subgroups of ##S_3## and find a 2-Sylow subgroup and a 3-Sylow subgroup of ##S_4.##

2. The attempt at a solution

I got ##|S_3| = 6 = 2\dot\ 3## and ##|S_4| = 24 = 2^3\dot\ 3.## So ##S_3## has a a Sylow 2 subgroup of order 2 and a Sylow 3 subgroup of order 3. I am asked to find the three 2-Sylow subgroups of ##S_3## so since the 2-Sylow of ##S_3## has order 2 is it just the permutations ##(1 2), (1 3), (2 3)? ##

Those are the generators of the only subgroups of [itex]S_3[/itex] of order 2, so yes.
 
  • Like
Likes 1 person

FAQ: Finding 2-Sylow & 3-Sylow of ##S_4##

What is a Sylow subgroup?

A Sylow subgroup is a subgroup of a finite group that has the largest possible order for that subgroup. It is named after the mathematician Ludwig Sylow, who first studied these subgroups.

How do you find the 2-Sylow subgroup of ##S_4##?

To find the 2-Sylow subgroup of ##S_4##, we can use the Sylow theorems. First, we need to determine the number of Sylow 2-subgroups in ##S_4##, which is equivalent to finding the number of ways to partition 4 elements into 2-element subsets. This can be done using combinatorics, and we find that there are 3 Sylow 2-subgroups in ##S_4##. Then, we can use the Sylow theorems to show that these subgroups are conjugate to each other, and thus, they are isomorphic.

How do you find the 3-Sylow subgroup of ##S_4##?

Similar to finding the 2-Sylow subgroup, we can use the Sylow theorems to determine the number of 3-Sylow subgroups in ##S_4##, which is equivalent to finding the number of ways to partition 4 elements into 3-element subsets. We find that there is only 1 3-Sylow subgroup in ##S_4##. To find this subgroup, we can use the fact that the order of the 3-Sylow subgroup must divide the order of ##S_4##, and then we can use the Sylow theorems to show that this subgroup is normal in ##S_4##.

Can there be more than one Sylow subgroup of the same order?

Yes, there can be multiple Sylow subgroups of the same order in a finite group. This is because the Sylow theorems only guarantee the existence of at least one Sylow subgroup of a given order, but there can be multiple subgroups with the same order that satisfy the conditions of the Sylow theorems.

Why are Sylow subgroups important?

Sylow subgroups are important because they allow us to better understand the structure of finite groups. They provide a way to break down a group into smaller, more manageable subgroups, and they also have important applications in group theory and other branches of mathematics.

Similar threads

Normal group of order 60 isomorphic to A_5
Replies
6
Views
1K
No group of order 10,000 is simple
Replies
1
Views
1K
A detail in a proof about isomorphism classes of groups of order 21
Replies
5
Views
2K
How Many Sylow 2-Subgroups Are There in S4?
Replies
5
Views
13K
Subnormal p-Sylow Subgroup of Finite Group
Replies
9
Views
2K
Existence of group of order 12 (Sylow's theorem?)
Replies
5
Views
3K
Calculating Galois Group of extension
Replies
1
Views
1K
Sylow's Theorem Proof for Group of Order 35^3 | Validity Check
Replies
1
Views
2K
Understanding Sylow's First Theorem to Prime Power Subgroups
Replies
2
Views
1K
Generalization of Third Sylow Theorem
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top