Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that there exists a subgroup of $G$ with order $d$ or $n/d$?

Of course this does not hold in full generality. -- In particular one quickly finds that e.g. ${\rm PSL}(2,8)$, ${\rm PSL}(2,11)$, ${\rm PSL}(2,13)$, ${\rm SL}(2,11)$, ${\rm SL}(2,13)$, ${\rm PSL}(2,17)$, ${\rm A}_7$, ${\rm PSL}(2,19)$, ${\rm A}_5 \times {\rm A}_5$, etc. are counterexamples.

Though does the assertion hold for

  • solvable groups?

  • solvable groups with derived length less than a certain bound $> 2$?

  • groups whose order has at most 2 distinct prime divisors?

A quick computation with GAP shows that any counterexample must have order $\geq 192$.


closed as off-topic by Mark Sapir, Andrey Rekalo, David White, Derek Holt, David Roberts Sep 26 '13 at 8:18

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  • 2
    $\begingroup$ @TobiasKildetoft: This is much simpler. The OP got this problem as a homework in his/her algebra class. He/she decided to post it here instead of doing the problem on his/her own. Since we do not know OP's home University and the name of the instructor, we cannot send a formal complain. So the OP is safe. $\endgroup$ – Mark Sapir Sep 26 '13 at 7:38
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    $\begingroup$ OK you two, how quickly can you think of a counterexample!!!! $\endgroup$ – Derek Holt Sep 26 '13 at 7:46
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    $\begingroup$ @MarkSapir: While I agree with you that homework questions are off-topic on MO and should be removed, I think your tone is unnecessarily rude. Also, we don't know whether the question is actually the OP's homework. I have therefore chosen to try to improve the question. $\endgroup$ – Stefan Kohl Sep 26 '13 at 16:22
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    $\begingroup$ While Mark Sapir may have been a little rude, the fact remains that this question is not suitable for MO (it is suitable for MSE), and continuing to discuss it, attempting to improve it, etc, is just encouraging the abuse of this site. If the overall level of questions on MO keeps going down, then the experts who contribute to the discussions will start abandoning the site. $\endgroup$ – Derek Holt Sep 26 '13 at 18:24
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    $\begingroup$ @Mark Sapir : Are we in a tribunal? if so, this would not be very encouraging for doing mathematics. We may lose many young students, who can do great things. Encouraging people is better than judging theme. $\endgroup$ – Yassine Guerboussa Sep 27 '13 at 11:17