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Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

  • $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
  • For each maximal subgroup $M'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?

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I think that you must omit (or both) in the question, because if for all maximal subgroups this occur then clearly the frattini subgroup is non-trivial and so the answer is NO! –  majid arezoomand Dec 8 '12 at 20:59
@higwain: When you change the question so that one of the answers becomes incorrect, then you should do the change in a way that makes it clear and obvious that you changed the question. Here you have added the condition "at least one of its maximal subgroups isn't of prime order", which makes the answer that had already been posted by majid arezoomand look as if majid didn't read your question carefully enough; you have, in essence, made it so that instead of it looking like an omission on your part, it looks like an error on his part. –  Arturo Magidin Dec 8 '12 at 23:46
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2 Answers 2

The answer in general is "No". For example consider the dihedral group $D_{10}$ of order 10. Then the frattini subgroup of $D_{10}$ is trivial and has 6 maximal subgroups that one of them is of order 5 and the others are of order 2. Now clearly, the answer is "NO".

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Note: The OP has changed the question to add the condition that at least one maximal subgroup not have prime order. –  Arturo Magidin Dec 8 '12 at 23:47
I apologise for the edition that held in the question after majid arezoomand's answer! –  higwain Dec 9 '12 at 5:47
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It seems that there are some examples. Let $G=D_{10}\times S_2$. Then the frattini subgroup of $G$ is trivial and the maximal subgroups are (that can be check by GAP): $M_1=\langle(6,7),(1,2,3,4,5)\rangle$, $M_2=\langle (2,5)(3,4),(1,2,3,4,5)\rangle$, $M_3=\langle(2,5)(3,4)(6,7),(1,2,3,4,5)\rangle$, $M_4=\langle(6,7),(2,5)(3,4)\rangle$, $M_5=\langle(6,7),(1,4)(2,3)\rangle$, $M_6=\langle(6,7),(1,2)(3,5)\rangle$, $M_7=\langle(6,7),(1,5)(2,4)\rangle$, and $M_8=\langle(6,7),(1,3)(4,5)\rangle$. Now consider $H_1=\langle(6,7)\rangle$ and $H_2=\langle(1,2,3,4,5)\rangle$.

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I think that the classification of such groups is a good problem –  majid arezoomand Dec 9 '12 at 10:06
@ majid arezoomand! many thanks for your examples, I think so! –  higwain Jan 22 '13 at 17:53
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