- #1
TheForumLord
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Homework Statement
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.
Homework Equations
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 really lame and I need to send a lot of excercices 'till next week so I really need your help :(
TNX to all the helpers!