• Support PF! Buy your school textbooks, materials and every day products Here!

Order of an element in a group

  • Thread starter missavvy
  • Start date
  • #1
82
0
Hey guys!

I'm having some trouble trying to solve this question.. Any advice/help is appreciated!

Homework Statement



Suppose that a and b belong to a group such that:
|a| = 4, |b| = 2, and suppose a3b=ba
Find the order of ab.

Homework Equations





The Attempt at a Solution


So I am unsure of which theorems I should be looking at... but anyways...... my horrible attempt:
so a,b are in G.
|ab| = (ab)n = 1

Let n = order(ab), anbn = 1 --> an=b-n

I don't know if I should continue from this point because I don't know how exactly I would go about finding the order.

I think |ab| = |ba|, yes or no?
If it does then I'm guessing I can use that to find the order of a3b?
Any hints or suggestions on how?

Thank you.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
(ab)^n and a^n*b^n are not the same thing. Clearly your group isn't abelian. Hint: (ab)(ab)=a(ba)b.
 
  • #3
82
0
Hey, thanks for the reply.
So is this a matter of manipulating a and b ?

For your hint, I'm just wondering why is it (ab)(ab)?
I see that a(ba)b = a^4b^2, which we have as the original orders, but why (ab)^2 and not some other number n?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Hey, thanks for the reply.
So is this a matter of manipulating a and b ?

For your hint, I'm just wondering why is it (ab)(ab)?
I see that a(ba)b = a^4b^2, which we have as the original orders, but why (ab)^2 and not some other number n?
Just because I noticed (ab)^2 worked. You can probably figure the order can't be too high with a relation like ba=a^3*b. You could try it for other n. You know n=4 will also work. The given relation tell you how to commute a and b. I.e. how to turn a 'ba' type expression into an 'ab' type expression.
 

Related Threads for: Order of an element in a group

  • Last Post
2
Replies
34
Views
1K
  • Last Post
Replies
1
Views
694
  • Last Post
Replies
8
Views
1K
Replies
3
Views
8K
Replies
1
Views
273
  • Last Post
Replies
7
Views
1K
Replies
2
Views
981
Replies
5
Views
3K
Top