If G is an infinite planar group (it means that it has a generating subset C such that Cay (S, C) is a planar graph) and H is a normal subgroup of it, I would be very grateful if somebody helps me and tell me "is G/H a planar group?"
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$\begingroup$ The usual terminology is that a group is planar if its subgroup lattice is planar. $\endgroup$– Igor RivinCommented Oct 1, 2014 at 14:31
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$\begingroup$ Thanks This is Meschke's definition of planar groups. You are righr! I should have remarked it. There are two definitions of planar groups. That is why I wrote the definition. I need this version. $\endgroup$– khersCommented Oct 1, 2014 at 14:35
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2 Answers
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The dihedral groups can be viewed as the set of all functions of the form $x\mapsto\pm x+c$ acting either on $\mathbb{Z}$ or on $\mathbb{Z}/n\mathbb{Z}$. The images of the infinite dihedral group are either finite dihedral groups. Taking as generators $x\mapsto x+1$ and $x\mapsto -x$ you get for the dihedral group of order $2n$ a planar Cayley graph consisting of two circles of length $n$.
Hence the infinite dihedral group is an example of an infinite non-abelian group, which has only planar images. I tried to find a more interesting example, but failed.
answered Oct 1, 2014 at 15:38
Jan-Christoph Schlage-PuchtaJan-Christoph Schlage-Puchta
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$\begingroup$ Thank you very very much I have the counter example that I need now. Again thanks so much to both of you $\endgroup$– khersCommented Oct 1, 2014 at 17:21
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With your definition, a free group is planar (since its Cayley graph is a tree), so if your conjecture were correct, then every finite generated group would be planar. This is obviously false (though no counterexample leaps to mind just now; presumably $PSL(n, 5)$ should contain $K_5$s for some $n$).
answered Oct 1, 2014 at 14:35
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$\begingroup$ Thanks so much! You are right again. I checked finite abelian groups and I infinite ones. But I completely forget the free groups. Do you know an infinite non-abelian planar group which has this property? $\endgroup$– khersCommented Oct 1, 2014 at 14:48
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$\begingroup$ @khers you mean: is there an infinite non-abelian planar group such that every quotient is planar? $\endgroup$ Commented Oct 1, 2014 at 14:52
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$\begingroup$ yes exactly Even now you helped me very much. Thanks again $\endgroup$– khersCommented Oct 1, 2014 at 14:58