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

Group Theory (Abstract algebra manipulations)

  • Thread starter pbxed
  • Start date
  • #1
14
0

Homework Statement


Let a,b,c,d be elements of a group G and let ab = c, bc = d, cd = a, da = b. Examine the expression da^2b and first derive an expression for b in powers of a. Then express c and d in powers of a. Show that a^5 = e (identity element)


Homework Equations





The Attempt at a Solution


So ive been banging me head against this one for a while now. I feel like its something simple and obvious but i cant figure it out. I have an expression for b:

da^2b = daab = (da)(ab) = bc = d

d^-1*daab=d^-1*d
eaab = e
b = a^-2

I cant figure out c or d, well i should say I have worked out values for c and d but know them to be wrong because I can't show a^5 = e. (and also I have the answers to this question)

Thanks
 

Answers and Replies

  • #2
22,097
3,278

Homework Statement


Let a,b,c,d be elements of a group G and let ab = c, bc = d, cd = a, da = b. Examine the expression da^2b and first derive an expression for b in powers of a. Then express c and d in powers of a. Show that a^5 = e (identity element)


Homework Equations





The Attempt at a Solution


So ive been banging me head against this one for a while now. I feel like its something simple and obvious but i cant figure it out. I have an expression for b:

da^2b = daab = (da)(ab) = bc = d

d^-1*daab=d^-1*d
eaab = e
b = a^-2

I cant figure out c or d, well i should say I have worked out values for c and d but know them to be wrong because I can't show a^5 = e. (and also I have the answers to this question)

Thanks
You know c=ab. So what happens if you substitute b=a-2 in that equation?
 
  • #3
14
0
Well c = a*a^-2
c=a^-1 right?

Thats the value I got but when I look at the solution its wrong.

Supposedly c = b^-2 = a^4
 
  • #4
22,097
3,278
Indeed, that is what we want to prove. You already know that [itex]c=a^{-1}[/itex]. And what we want to prove is that [itex]a^{-1}=a^{4}[/itex]. But to prove that, you will want to calculate d first...
 
  • #5
14
0
Oh, I just worked it out. God I feel dumb.

Thx for your help micromass :)
 

Related Threads for: Group Theory (Abstract algebra manipulations)

Replies
1
Views
424
Replies
4
Views
3K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
0
Views
1K
Top