Group Theory (Abstract algebra manipulations)

[pbxed]

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 I've been banging me head against this one for a while now. I feel like its something simple and obvious but i can't 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 can't 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

 

[micromass]

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 I've been banging me head against this one for a while now. I feel like its something simple and obvious but i can't 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 can't 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?

 

[pbxed]

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

 

[micromass]

Indeed, that is what we want to prove. You already know that c=a^{-1}. And what we want to prove is that a^{-1}=a^{4}. But to prove that, you will want to calculate d first...

 

[pbxed]

Oh, I just worked it out. God I feel dumb.

Thx for your help micromass :)