# Earthling

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 Apr19 comment Automorphism group of Lie algebra $\mathfrak{g\oplus g}$ @MarianoSuárez-Alvarez I gave up trying to write it as a quotient and edited the question. Apr19 comment Easy problems on continuity You sure? What happens with (i) when $x\to 0$? Google "cos(1/x)", Google seems to have a plotting function now. Apr18 comment Automorphism group of Lie algebra $\mathfrak{g\oplus g}$ True. This seems to be less trivial than I imagined. Is $\operatorname{Aut}(\mathfrak{g\oplus g})\cap\operatorname{Aut}(\Delta\mathfrak g)$ normal in $\operatorname{Aut}(\mathfrak{g\oplus g})$? Apr18 comment Automorphism group of Lie algebra $\mathfrak{g\oplus g}$ Yes, I mean $\operatorname{Aut}(\mathfrak{g\oplus g)/\operatorname{Aut}(\Delta g})$. Apr18 comment Automorphisms of direct sum of Lie algebras Back to the blackboard, then. Thank you for your comments. Apr18 comment Automorphisms of direct sum of Lie algebras @MarianoSuárez-Alvarez I see. My bad. I found this though. Apr18 comment Choosing central generator for nilpotent group generated by 3 elements I found a comment on this question on MO, which claims that one generator can be chosen to be central. However, I do not understand the reasoning by which the commenter deduced this fact. Could there be a simple proof for the case when the number of generators is 3? Apr18 comment Choosing central generator for nilpotent group generated by 3 elements Thanks. I guess that means that even in fairly simple cases (nilpotence class 2, 3 generators, torsion-free), these conditions imply very little about the group. I also guess it's a measure of my mathematical ignorance that makes me state overly strong conjectures. Do you have any idea what happens if I add the condition that the centre be generated by one element, i.e. infinite cyclic? Apr17 comment Choosing central generator for nilpotent group generated by 3 elements Yes, really really sorry I didn't remember to put it into the question. By 2-step I mean that $[a,b]$ is in the centre of $G$, i.e. trivial group "in two steps". Apr17 comment Choosing central generator for nilpotent group generated by 3 elements Thanks, that's a neat counter-example. I would like to add the condition "torsion-free" to $G$. Apr11 comment How to show that $\mathrm{Sym}_{n\times n}(\Bbb{R})$ and $\mathrm{Skew}_{n\times n}(\Bbb{R})$ are subspaces of $\mathrm{M}_{n\times n}(\Bbb{R})$ Sorry I hadn't finished reading your proof, which seems complete to me. Apr11 comment How many discrete subgroups does the Heisenberg group have? @MarianoSuárez-Alvarez I think the family is closed under conjugation and under taking subgroups. Apr11 comment When are homology groups free abelian groups? Look up homology of real projective space. Aug24 comment Determine topological properties of a space of matrices Got it, sorry for the noise. Aug23 comment Determine topological properties of a space of matrices I have already accepted your answer. Still, even after taking a closer look, I can't even reason backwards on how you deduce that $\det X\neq\det (X+I)$. What did you use to see this? Aug20 comment Determine topological properties of a space of matrices Nice, thank you. The first factor is $\mathrm{GL}_2\mathbb R$. Any ideas for what the homotopy type of $\{X\in M_2\mathbb R\mid\mathrm{tr}\,X\neq -1\}$ is? Aug20 comment Determine topological properties of a space of matrices Connectedness, homotopy groups, homology groups, cohomology groups. These would all follow from general theory, if one could write down explicitly, what this space is, as for the case when one just imposes the second condition. Jul26 comment Structures on torus That is much more than I expected from any answer. Thank you so much! Jul26 comment $\mathrm{det}(A-B)\neq0$ if and only if $\mathrm{det}(A+B)\neq0$? @all: I tried to delete as soon as I hit the submit button, but at first there was no delete button, and by the time it appeared, there was already an answer, and the site told me I would have to flag an admin if I wanted to vote for deletion. It is quite curious, however, how much response an embarrassingly bad question elicits. Even worse, I have actually gained reputation with this question, because downvotes have less impact than upvotes. I was doing calculations which seemed to imply my question and was too quick to ask. My sincere apologies to all. Jul26 comment Structures on torus Thanks. If I understand it correctly, points in the Teichmüller space for the torus correspond to complex structures, but the various metrics on the Teichmüller space don't correspond to metrics on the torus... How about the Riemannian metric inherited from $\mathbb R^2$. This one should vary with the lattice, I would assume. I just don't know how...