# Can a normalizer be described by generators and relations?

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.)

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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 $|M|=|S_5|/|N|$.