Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Show that the group with presentation $$\langle a,b| aba^{-1}=b^n, b=(ba)^2\rangle$$ is a cyclic group generated by $a$ and determine its order.

share|improve this question
    
What have you tried? If this is homework, tag it as such. –  lhf Oct 1 '12 at 14:35

2 Answers 2

Hint: $b = (ba)^2$ implies $b = baba \Rightarrow aba = e \Rightarrow b = a^{-2}$.

share|improve this answer
1  
Could you say more about how this determinates the order of $a$ ? –  Ragib Zaman Oct 1 '12 at 15:05
2  
@Ragib: I want to leave some work for the OP to do. –  Brandon Carter Oct 1 '12 at 15:13

$$b=(ba)^2=baba=\Longrightarrow aba=1=aba^{-1}b^{-n}\Longrightarrow a=a^{-1}b^{-n}\Longrightarrow$$

$$1=aba=(a^{-1}b^{-n})ba=a^{-1}b^{-n+1}a\Longrightarrow b^{n-1}=1\Longrightarrow b^n=b=a^{-2}$$

Can you take it from here?

share|improve this answer
1  
The last implication on the top row isn't correct. Should be $a^{-1}b^{-n} = a$. –  Brandon Carter Oct 1 '12 at 14:29
    
It's already been taken care of. thanks –  DonAntonio Oct 1 '12 at 14:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.