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.

Suppose that $\alpha,\beta \in S_4, \,\alpha (3) = 4,\, \alpha\beta = (3412), \,\beta\alpha = (3241).\;$ Find $\alpha\,$ and $\, \beta$.

My attempt:

Ok so I know

$\alpha(3) = 4$ and $\alpha(\beta(3))=4$ so $\beta(3)=3$

$\beta(3)=3$ and $\beta(\alpha(1))=3$ so $\alpha(1) = 3$

$\alpha(1) = 3$ and $\alpha(\beta(2))=4$ so $\beta(2)=1$

$\beta(2)=1$ and $\beta(\alpha(4))=1$ so $\alpha(4) =2$

$\alpha(4) = 2$ and $\alpha(\beta(1))=2$ so $\beta(1)=4$

But then I just get back to $\alpha(3) = 4$, and am stuck.

Any help would be appreciated. Thanks.

share|improve this question
1  
What is $i$? Is $\alpha(3)$ equal to $3$ as in the title or $4$ as in the text? –  Jim Feb 10 '13 at 19:11
    
my mistake, $\alpha(3) = 4$ –  bobdylan Feb 10 '13 at 19:14
1  
Suggest writing down the cycles for $\alpha$ and $\beta$ - what are you left with, and can you fill the gaps? –  Mark Bennet Feb 10 '13 at 19:19
add comment

2 Answers

up vote 4 down vote accepted

Every permutation is a bijection, including $\alpha$ and $\beta$.

Hence, since you've found $\beta(1) = 4, \beta(2) = 1, \beta (3) = 3,$ that leaves only the possibility that $\beta(4)=2$.

Similarly, $\alpha(1) = 3, \alpha(2) = ?, \alpha(3) = 4, \alpha(4) = 2 \implies \alpha(2) = 1$

So write the cycles: $\alpha, \beta$ and you're done:

$$\alpha: 1\to 3, \;3 \to 4,\; 4\to 2,\; 2 \to 1 \;\implies\; \alpha = (1342)$$ $$ \beta: 1 \to 4, \;4 \to 2, \;2 \to 1;\; \,3\to 3 \;\implies \;\beta = (142)(3) = (142)$$

share|improve this answer
add comment

$\beta$ is a bijection. you already found that $\beta(1)=4,\beta(2)=1,\beta(3)=3$. Thus, $\beta(4)=2$. Now that you find $\beta$ you can get $\alpha=(3412)\beta^{-1}$

share|improve this answer
    
+1: Or, for that matter, the OP could use similar argument to determine $\alpha(2)$, since the other three function values have been determined. –  Cameron Buie Feb 10 '13 at 19:39
    
@Cameron Buie Yes thats less work –  Amr Feb 10 '13 at 19:41
add comment

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.