Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to use generators and relations here.

Let M ≤ S_5 be the subgroup generated by two transpositions t_1= (12) and t_2= (34).

Let N = {g ∈S_5| gMg^(-1) = M} be the normalizer of M in S_5.

How should I describe N by generators and relations?

How should I show that N is a semidirect product of two Abelian groups?

How to compute |N|?

How many subgroups conjugate to M are there in S_5 ? Why?

(I think Sylow's theorems should be used here.)

share|cite|improve this question
Note that if $g$ normalizes $M$, then $g$ cannot move the point $5$ (why not?), so you are really doing this in $S_4$... – user641 Nov 9 '12 at 1:46

Hint: Show that $M=\{id,(12),(34),(12)(34)\}$. What does a conjugation $gsg^{-1}$ mean for a cycle $s\in S_5$? Find some elements of $N$, then try to describe all elements.

If you have $|N|$, for the last question, consider the orbit of $M$ under the action of $G$ by conjugation on the set of subgroups: $$\langle g, H\rangle\mapsto gHg^{-1}$$ Then $N$ is exactly the stabilizer of $M$, and show that for any elements $x,y\in G$, we have $xMx^{-1} = yMy^{-1} \implies x^{-1}y\in N$, and conclude that $|M|=|S_5|/|N|$.

share|cite|improve this answer
N={id,(12),(34)}? – Jack Nov 9 '12 at 2:10
Or N={id,(12),(34),(12)(34)}? – Jack Nov 9 '12 at 2:13
What is G ??????? – Jack Nov 9 '12 at 2:32

Your Answer


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.