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} )-I've managed to prove this one...
Can't understand how to prove this one:
B. Prove that the normalizer of H is a 2-sylow group of S4.
TNX to all the helpers!
I didn't completely understand...
The equality you've wrote is obvious... By the definition, it means that (12)(34) is in N[Sn](H) ofcourse...
but why it means that 4|o(C(H))? And why from here we get that it's a sylow-p group? It can has an order 12 also...
TNX
Originally Posted by WannaBe
That was a typo: it should have been, and it follows at once from Lagrange's theorem and the fact that for any subgroup H of any group G we always have
.
Finally, no:can't be all
(which is the only subgroup of order 12 in
)
Tonio
Well, this depends on what you consider "easy", but the following isn't, imo, hard:
Lemma: If, then either
or else exactly half the elements of
are even permutations (and thus, of course,
)
Proof: (Hint) If, define a map from the set of odd permutations of
to
the set of even permutations of H, by
, and check it is
...
From this lemma it follows at once what we want.
Tonio
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