What is the Significance of the Orbit of P in Sylow's Theorems?

  • Thread starter Thread starter mathsss2
  • Start date Start date
  • Tags Tags
    Subgroup
Click For Summary
The discussion focuses on the significance of the orbit of a p-Sylow subgroup P in relation to Sylow's Theorems. It highlights that if M is a subgroup of G containing the normalizer N_G(P), then the index [G:M] is congruent to 1 modulo p. The action of G on P by conjugation leads to the permutation of subgroups, with the orbit of P containing kp+1 subgroups. This relationship is crucial for understanding the structure of finite groups and the implications of Sylow's Theorems. The exploration of these concepts reveals deeper insights into the properties of group actions and subgroup dynamics.
mathsss2
Messages
38
Reaction score
0
Let p be a prime, G a finite group, and P a p-Sylow subgroup of G. Let M be any subgroup of G which contains N_G(P). Prove that [G:M]\equiv 1 (mod p). (Hint: look carefully at Sylow's Theorems.)
 
Physics news on Phys.org
since the word N = normalizer occurs, one is led to look at the action on G on p by conjugation. G permutes subgroups of G and we look at the orbit of P. this orbit contains kp+1 subgroups, so the theorem holds if N = M. then what?
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
View full post »

Similar threads

Undergrad What is the importance of p-Sylow subgroups in understanding group structure?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Normal subgroups (Sylow Theorem)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Determining the number of conjugacy classes in a p-group
  • · Replies 6 ·
Replies
6
Views
2K
MHB Why Can't a Proper Normal Subgroup Contain a Sylow Normalizer in a Group?
  • · Replies 1 ·
Replies
1
Views
1K
MHB G contains a normal p-Sylow subgroup
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Is G simple?Is G simple? A different approach using Sylow theorems
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad First Sylow Theorem: Group of Order ##p^k## & Cyclic Groups
  • · Replies 1 ·
Replies
1
Views
2K
MHB Is \( f(P) \) a \( p \)-Sylow Subgroup of \( H \)?
  • · Replies 3 ·
Replies
3
Views
2K
A lemma in proving Sylow's theorem
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Sylow subgroup of some factor group
  • · Replies 3 ·
Replies
3
Views
2K