Proving A5 has No Normal Subgroups: Conjugacy Classes Approach

  • Thread starter Thread starter Obraz35
  • Start date Start date
  • Tags Tags
    Groups
Click For Summary
SUMMARY

The discussion focuses on proving that the alternating group A5 has no normal subgroups other than itself and the trivial subgroup {e}. The approach suggested involves analyzing the conjugacy classes of A5, as a normal subgroup must be a union of these classes. The user seeks guidance on determining the conjugacy classes, with a reference to the conjugacy classes of elements in the symmetric group S5 as a potential starting point.

PREREQUISITES
  • Understanding of group theory, specifically the properties of normal subgroups.
  • Familiarity with the concepts of conjugacy classes in group theory.
  • Knowledge of the structure and properties of the alternating group A5.
  • Basic understanding of the symmetric group S5 and its relation to A5.
NEXT STEPS
  • Research the conjugacy classes of the symmetric group S5.
  • Study the properties of the alternating group A5 in detail.
  • Learn about centralizers and their role in group theory.
  • Explore proofs regarding normal subgroups in finite groups.
USEFUL FOR

Mathematicians, particularly those studying group theory, graduate students in abstract algebra, and anyone interested in the properties of the alternating group A5.

Obraz35
Messages
29
Reaction score
0

Homework Statement


I am interested in proving that A5 has no normal subgroups except itself and {e}.


The Attempt at a Solution


Some proofs that I have seen use centralizers to do this, but since I haven't gone through that yet I think there should be some say to do it without them.

My approach would be to find the conjugacy classes of A5 and use their orders to show that there cannot be a normal subgroup in A5 since a normal subgroup is a union of conjugate classes.
But my main problem is how I should go about finding the conjugacy classes.

Thanks for your help.
 
Physics news on Phys.org
There is an identical thread to this in this forum. The advice there is: do you know the conjugacy classes of elements in S_5?
 

Similar threads

Graduate Decomposition into irreps of compact Lie group
  • · Replies 4 ·
Replies
4
Views
1K
Abstract Algebra - Group of Order 12 with Conjugacy Class of Order 4
  • · Replies 2 ·
Replies
2
Views
5K
Undergrad Determining the number of conjugacy classes in a p-group
  • · Replies 6 ·
Replies
6
Views
3K
Having all subgroups normal is isomorphism invariant
  • · Replies 1 ·
Replies
1
Views
2K
Conjugacy and Stabilizers in Group Actions
  • · Replies 1 ·
Replies
1
Views
1K
Using class equation to show that intersection is nontrivial
  • · Replies 1 ·
Replies
1
Views
2K
Showing that every finite group has a composition series
  • · Replies 3 ·
Replies
3
Views
3K
What Are the Conjugacy Classes of the Alternating Group A4?
  • · Replies 1 ·
Replies
1
Views
21K
Proving Non-Simplicity in Finite Groups with Two-Element Conjugacy Class
  • · Replies 1 ·
Replies
1
Views
2K
Proving Normality of [0] in Z/3Z Quotient Group
  • · Replies 4 ·
Replies
4
Views
4K