(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In the permutations group S4, let H be the cyclic group the is generated by the cycle

(1 2 3 4).

A. Prove that the centralizer C(H) of H is excatly H ( C(H)={g in G|gh=hg for every h in H} )

B. Prove that the normalizer of H is a 2-sylow group of S4.

2. Relevant equations

3. The attempt at a solution

Well, there's a quick calculation of H:

(1234)(1234)=(13)(24)

(13)(24)(1234)=(1 4 3 2)

(1432)(1234)=(1)(2)(3)(4)

Hence: H={(1234),(13)(24),(1432),1}

The order of the group is 4. Hence [S4]=6...

Since it's a cyclic group, it's abelian. Which means H is a sub group of C(H)...

How can I prove that C(H) is in H? Or how excatly can I prove the 1st statement?

About 2, I actually have no clue.... My abilities in sylow&homomorphzm are realy lame and I need to send a lot of excercices 'till next week so I realy need your help :(

TNX to all the helpers!

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# Homework Help: Abstract-Sylow Theorems

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