Centralizers and elements of odd order

  • Thread starter Thread starter wakko101
  • Start date Start date
  • Tags Tags
    Elements
Click For Summary

Homework Help Overview

The discussion revolves around a proof involving group theory, specifically focusing on the properties of centralizers and elements of odd order. The original poster seeks clarification on the implications of a specific equation involving elements that centralize each other.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the relationship between the commutation of elements and the implications of odd order on centralization. There is an examination of the algebraic manipulation of expressions involving powers of elements.

Discussion Status

Some participants express understanding of the reasoning presented, while others reiterate the steps taken to arrive at the conclusion. The discussion reflects a collaborative effort to clarify the proof without reaching a definitive consensus on the final interpretation.

Contextual Notes

Participants are working within the constraints of group theory and the properties of elements of odd order, with specific attention to the implications of given equations and their algebraic transformations.

wakko101
Messages
61
Reaction score
0
I was going over a proof in which the following is given:

if abb=bba and b is of odd order, then ab=ba (i.e. if b^2 centralizes a then so does b)

I'm not sure why this is so. Any clarification would be appreciated.

Cheers,
W. =)
 
Physics news on Phys.org
b^2 commutes with a. If b has odd order then b^(2n+1)=e for some n. So b^(2n+1)a=a(b^(2n+1)). Do you see it now?
 
I believe so...
b^(2n)ba = ab^(2n)b
b^2n(ba) = b^2n(ab) since abb = bba
ba = ab

If that's it, then cheers! =)
 
wakko101 said:
I believe so...
b^(2n)ba = ab^(2n)b
b^2n(ba) = b^2n(ab) since abb = bba
ba = ab

If that's it, then cheers! =)

Cheers!
 

Similar threads

  • · Replies 5 ·
Replies
5
Views
1K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 7 ·
Replies
7
Views
6K