3
$\begingroup$

Let $G_1, G_2 $ and $G_3$ be subgroups of $G$ with finite indexes. Suppose that there are $x_1,x_2,x_3\in G$ such that

  • $G_1x_1\cap G_2x_2=G_2x_2\cap G_3x_3=G_1x_1\cap G_3x_3=\emptyset$,
  • $G_1G_2x_2\cup G_1G_3x_3=G_2x_2\cup G_3x_3$,
  • $G_2G_1x_1\cup G_2G_3x_3=G_1x_1\cup G_3x_3$, and
  • $G_3G_1x_1\cup G_3G_2x_2=G_1x_1\cup G_2x_2$.

Must it be that at least two of $|G:G_1|$, $|G:G_2|$, $|G:G_3|$ are equal?

$\endgroup$
5
  • 1
    $\begingroup$ Where $x_1$, $x_2$, and $x_3$ are what? Arbitrary elements of $G$? $\endgroup$ Dec 21, 2010 at 2:03
  • 4
    $\begingroup$ Also, it would help to have some of the motivation for this question. Where did it come up? $\endgroup$ Dec 21, 2010 at 2:04
  • $\begingroup$ yes, $x_i$ is arbitrary. $\endgroup$
    – Rulin
    Dec 21, 2010 at 2:14
  • $\begingroup$ Rulin, please let me know if I edited incorrectly. $\endgroup$ Dec 21, 2010 at 2:44
  • $\begingroup$ very good, correctly. $\endgroup$
    – Rulin
    Dec 21, 2010 at 3:16

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.