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relation between p-sylow subgroups

If I have two p-sylow groups $P_{1}, P_{2}$ can I then be sure that

$x\in P_{1} \Rightarrow x \notin P_{2}$?

That is to say that all the elements in $P_{1}$ are distinct from the elements in $P_{2}$?

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marked as duplicate by wentaway, Chris Eagle, Leonid Kovalev, Zhen Lin, t.b. Aug 16 '12 at 11:36

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2  
Well, identity element? –  user21436 Mar 20 '12 at 11:04
    
Exact Duplicate, Look at my answer and other linked questions...: math.stackexchange.com/q/111730/21436 –  user21436 Mar 20 '12 at 11:07

2 Answers 2

up vote 3 down vote accepted

No. An easy counterexample would be the direct product $G=P\times N$, where $P$ is a $p$-group, and $N$ is a group with several Sylow $p$-subgroups. Then all the Sylow $p$-subgroups of $G$ are of the form $P\times P'$, where $P'$ is a Sylow $p$-subgroup of $N$. All these intersect non-trivially at $P$.

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Nice Answer. In the question, I link there, I have been hand wavy about an example. But, good example. +1! –  user21436 Mar 20 '12 at 11:08

The intersection of all the Sylow $p$-subgroups is called the $p$-core of $G$ and is the biggest normal $p$-subgroup of $G$. In other words, if you have a normal $p$-subgroup in $G$, then this subgroup is contained in all the Sylows. Easy example: dihedral group of order 20. The centre is of order 2 and is contained in every Sylow 2-subgroup.

This is not the only way that two Sylows can intersect. The intersection of two Sylows can be non-trivial even if the $p$-core is trivial. For example in $A_{10}$, the Sylow 5-subgroups are each generated by 2 disjoint 5-cycles. The groups $\langle (1,2,3,4,5), (6,7,8,9,10)\rangle$ and $\langle (1,2,3,4,5), (6,7,8,10,9)\rangle$ intersect in a cyclic subgroup of order 5. But the 5-core is trivial, since $A_{10}$ is simple.

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Nice Answer, +1. :-) –  user21436 Mar 20 '12 at 14:13
    
+1 Good to see an example, where the $p$-core is trivial. –  Jyrki Lahtonen Mar 20 '12 at 20:07

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