Conjugating Sylow 3-subgroups in S_4

  • Thread starter Thread starter ehrenfest
  • Start date Start date
  • Tags Tags
    Theory
Click For Summary
SUMMARY

The discussion focuses on finding and demonstrating the conjugacy of all Sylow 3-subgroups in the symmetric group S_4. The Sylow 3-subgroups identified are generated by the elements <(1,2,3)> and <(2,3,4)>. The key challenge is determining the conjugating elements, which can be efficiently approached by leveraging symmetry rather than exhaustively testing all 24 elements of S_4. The solution emphasizes the importance of careful selection of elements to minimize the number of trials needed to establish conjugacy.

PREREQUISITES
  • Understanding of Sylow theorems and their applications in group theory.
  • Familiarity with symmetric groups, specifically S_4.
  • Knowledge of group actions and conjugation in the context of abstract algebra.
  • Basic proficiency in permutation notation and cycle notation.
NEXT STEPS
  • Study the properties of Sylow subgroups in various groups, focusing on Sylow 3-subgroups.
  • Learn about conjugacy classes in symmetric groups, particularly S_4.
  • Explore the application of group actions to simplify the process of finding conjugating elements.
  • Investigate the general case of conjugation in groups beyond S_4, applying Sylow theorems.
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, group theory enthusiasts, and educators looking to deepen their understanding of Sylow subgroups and conjugacy in symmetric groups.

ehrenfest
Messages
2,001
Reaction score
1
[SOLVED] Sylow theory

Homework Statement


Find all Sylow 3-subgroups of S_4 and demonstrate that they are all conjugate.


Homework Equations





The Attempt at a Solution


I found all the Sylow 3-subgroups, but I am having trouble finding the element that conjugates them. For example, how do you find the element conjugates that <(1,2,3)> and <(2,3,4)>? I could just try all the elements, but there are 24, so that is probably a bad idea...
 
Physics news on Phys.org
well, you can just try all elements, leave a few by symmetry.

But you'd only have to do them all once, because if you did it correct you'll see all the sylow subgroups emerge =)

If choose carefully you'll probably only have to do a bit more than there are subgroups in the class.
 
Last edited:
You could also try to solve the general case:
[tex]p (1, 2, 3) p^{-1} = (a,b,c)[/tex]
with
[tex]a\neq b \neq c[/itex][/tex]
 

Similar threads

Normal group of order 60 isomorphic to A_5
  • · Replies 6 ·
Replies
6
Views
2K
No group of order 10,000 is simple
  • · Replies 1 ·
Replies
1
Views
2K
Finding 2-Sylow & 3-Sylow of ##S_4##
  • · Replies 1 ·
Replies
1
Views
2K
Sylow's Theorem Proof for Group of Order 35^3 | Validity Check
  • · Replies 1 ·
Replies
1
Views
2K
Showing that a subgroup of Sym(4) is isomorphic to D_8
  • · Replies 1 ·
Replies
1
Views
2K
Subnormal p-Sylow Subgroup of Finite Group
  • · Replies 9 ·
Replies
9
Views
2K
A detail in a proof about isomorphism classes of groups of order 21
  • · Replies 5 ·
Replies
5
Views
2K
Normal subgroup generated by a subset A
  • · Replies 15 ·
Replies
15
Views
3K
Existence of group of order 12 (Sylow's theorem?)
  • · Replies 5 ·
Replies
5
Views
4K
What Determines the Normalizer of a Sylow p-Subgroup in Sym(p)?
  • · Replies 1 ·
Replies
1
Views
4K