A Sylow $p$-subgroup can be a subset of union of the rest of the Sylow $p$-subgroups?

Question

Let G be a finite group and assume that number of the Sylow $p$-subgroups is more than one.

My question is this: "Can a Sylow $p$-subgroup be a subset of the union of the rest of the Sylow $p$-subgroups?"

I think this is impossible. Every Sylow $p$-subgroup contains some elements not contained in rest of the union, but I can not prove it nor find counterexample; if you show me one of them, I would be thankful.

Actually, counterexamples do exist. One example is $\operatorname{PSL}(2,11)$ with $p = 2$. I asked a similar question here — Mikko Korhonen, Nov 10 '13 at 21:57
thanks,it is very interesting for me since if one of them is subset of rest then it is true for all of them (by conjugation).I can't visualize the picture of sylow-p groups in that case and I beleive they may satisfy more beautiful properties. — mesel, Nov 11 '13 at 20:39

score 0 · Answer 1 · answered Nov 10 '13 at 21:56

0

This is not true in general: take $G=S_3$, then $Syl_2(G)=\{\langle (1 2) \rangle,\langle(1 3)\rangle,\langle(2 3)\rangle\}$. Clearly each of these subgroups is not contained in the union of the others.

answered Nov 10 '13 at 21:56

Nicky Hekster

49,281

A Sylow $p$-subgroup can be a subset of union of the rest of the Sylow $p$-subgroups?

1 Answers1