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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 $S_4$...2012-11-09

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