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.

a) Determine the smallest symmetric group $S_n$ that contains a subgroup isomorphic to H, generated by $x^4=y^3=1$, $xy=y^2x$.

b) Find a subgroup of $SL_2(F_5)$ that is isomorphic to that group.

My first step was to say that $n$ is at least 4 so that $S_n$ contains an element of order 4. But I tried $S_4$ and it didn't quite work. After that I couldn't make much progress.

Any help appreciated.

share|improve this question
1  
My answer to this question can also be read as a hint for a): math.stackexchange.com/questions/233939/… –  Micah Nov 13 '12 at 2:27
    
@Micah Could you elaborate? Do you mean there is no such $S_n$? –  Benjamin Lu Nov 13 '12 at 4:25
    
$H$ is a finite group, so it certainly embeds in $S_n$ for some $n$ (by Cayley's theorem). I think if you experiment with some simple cycle types for $y$, and how it's possible for $x$ to conjugate $y$ appropriately while still having order $4$, it shouldn't be too tough to come up with the smallest possible $S_n$. –  Micah Nov 13 '12 at 4:29
    
@Micah is it $S_{12}$? –  Benjamin Lu Nov 13 '12 at 8:24
    
No it's not $S_{12}$. You know from an earlier question that $n>6$, so why not try $n=7$. –  Derek Holt Nov 13 '12 at 8:35
show 3 more comments

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.