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added the statement (here) that of all finite subgroups of $SU(2)$, $Q_8$ is a proper subgroup of the three exceptional ones.
Checking normality of this subgroup, I noticed that there is an issue with another item of the entry here, where it used to claim that a finite group is Hamiltonian precisely of it “contains a copy of $Q_8$”. But this can’t be, can it. I changed it to saying that every Hamiltonian group contains $Q_8$ as a subgroup, which I suppose is what was meant.
[edit: I see now that the statement that I changed back to was made already by Thomas Holder in rev 3, while the statement I removed was made by Thomas in rev 4. Thomas, if you see this, please let me know. ]
Thanks!
Just to double-check: Is it right that $Q_8 \subset 2T$ and $Q_8 \subset 2I$ are normal inclusions, but $Q_8 \subset 2 O$ is not normal (here)?
Isn’t it
binary octahedral group - not normal
binary tetrahedral group - normal
binary icosahedral group - not normal
Right, thanks. I have fixed it now. Also here.
Is the fact that the quaternion group $Q_8$, in its identification as the binary dihedral group $2 D_4$, carries the ADE-lable D4, the one corresponding to triality, usefully related to it being a proper subgroup precisely of the three exceptional finite subgroups of SU(2)?
I have reorganised things a bit to introduce the dicyclic and generalised quaternion groups earlier. I am wondering if the title is a good one for the parge as it now is. I think I have used the notation $Div_n$ correctly but if someone could check as there are different traditions in the literature and I may have got confused.
added graphics (here) of the subgroup lattices of the first few generalizes quaternion groups.
It looks like for all generalized quaternion groups we have that the minimal non-trivial subgroups are in the center (in fact, there is one non-trivial minimal subgroup equal to the center), and that they are the only non-abelian finite subgroups of $SU(2)$ with this property. Is this true?
Added pointer to Prop, 3.5 in
for the group cohomology
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