I am trying to solve a problem involving automorphism of a group.There needs the following argument: a p subgroup is contained in a sylow p subgroup.Is it true?I can't prove it,may be it is elementary.Plz help me. I tried it producting with a sylow p subgroup,although its order is p power bt it may not be a group since there is no normal argument is given.
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If a Sylow $p$-subgroup $P$ of group $G$ is by definition a maximal $p$-subgroup (i.e. if $P\leq Q\leq G$ and $Q$ is a $p$-subgroup then $P=Q$) then we can prove it like this:
If $H$ is a $p$-subgroup then there are $2$ possibilities: $H$ is a maximal $p$-subgroup (then $H\leq H$ tells us that $H$ is contained in a maximal $p$-subgroup) or we can find a $p$-subgroup $H'$ that properly contains $H$. In the second case we can repeat that for $H'$ and if $G$ is finite then eventually we will arrive at a maximal $p$-subgroup that contains $H$. If $G$ is not finite then it can be shown by means of Zorn's lemma.
drhab
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How the maximal p-subgroups are sylow subgroup? – Panja Sep 13 '14 at 14:48
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In finite case can we prove it?is it true? – Panja Sep 13 '14 at 15:34
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What is your definition for a Sylow $p$-subgroup? It is indeed true that any $p$-subgroup is contained in a Sylow $p$-subgroup. – drhab Sep 13 '14 at 16:31
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|G|=p^m.n,p does not divide n then every subgroup of order p^m is sylow subgroup of G.If H subgroup of G with |H|=p^k where k strictly less than n then does there exit a sylow p-subgroup of G containing H?If exits then how? – Panja Sep 13 '14 at 16:52
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Have a look at the answers on this question. – drhab Sep 13 '14 at 17:48
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thanks @drhab..I saw the proof before bt forgot. – Panja Sep 13 '14 at 18:07