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8h
comment Why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$?
@Chuks: I already did. Finding a subgroup of index $n$ is equivalent to finding a transitive action on a set of size $n$ (given such an action, the subgroup is the stabilizer of any point). I gave such an action, namely the natural action on projective space.
9h
answered What *is* affine space?
11h
comment Why every finite non-abelian simple group of order $n$ has a proper subgroup of index at most $kn^{\frac{3}{7}}$?
@Chuks: I already found the subgroup; the computation of the size of the group itself is just to verify that the subgroup I found is large enough.
13h
comment Can a (Hausdorff) infinite group have only finitely many equivalence classes of irreducible unitary representations?
Do you mean to include topological groups? If so do you want some hypotheses on the topology? (Hausdorff is clearly necessary or else there are dumb counterexamples involving groups with the indiscrete topology.)
1d
comment What is the convention for the codimension of an empty set?
@Fred: the OP's question is about algebraic sets. In that setting every reasonable notion of dimension and codimension should agree and should be calculable from Zariski tangent spaces. There's no need to entertain strange pathologies here. In any case, if that was the point you actually wanted to make you could've done it in one comment instead of three.
1d
awarded  Good Answer
2d
awarded  Nice Answer
2d
comment What is the convention for the codimension of an empty set?
@Fred: that doesn't agree with the meaning of codimension that I'm familiar with.
2d
comment Monoids in Category Theory
@melston: the latter.
2d
comment Monoids in Category Theory
@melston: no, that's just another abuse of notation.
2d
answered Monoids in Category Theory
2d
comment Can one speak of a threefold (or other) symmetry of SU(3) and the Gell-Mann matrices?
Are you asking for symmetries of $SU(3)$ as a group, as a Lie algebra, or as a manifold?
2d
comment What is the convention for the codimension of an empty set?
@Fred: sure, but of course the dimension and the codimension should sum to the ambient dimension, so the two determine each other. When I say the dimension of the empty set should be $-\infty$ I'm also saying that its codimension in any nonempty variety should be $\infty$.
Jul
3
answered Degree of a map over a different ring in homology
Jul
2
comment What is the convention for the codimension of an empty set?
In general, the supremum over no elements of a poset is its smallest element, if that exists. Here the poset is $\mathbb{Z}$ with a smallest element $-\infty$ adjoined.
Jul
2
answered What is the convention for the codimension of an empty set?
Jul
2
comment Is there a connection between lattices in the sense of orders and lattices in the sense of groups?
"By chance" is a bit too strong. Both of them involve lattice patterns in the colloquial sense; think about $\mathbb{Z}^2$ and then rotate it 45 degrees.
Jul
2
comment Can we recover homology from cohomology
@mqx: I don't follow. $H^1(-, \mathbb{Z})$ is always torsion-free.
Jul
2
comment Galois groups of quintics
@Servaes: not sure off the top of my head. Probably it's known how to get Galois group the dihedral group in general. Again I would start by first trying to find a Galois extension with that Galois group and only then finding a polynomial (by finding a primitive element). A dihedral extension is a cyclic extension of a quadratic extension, so probably Kronecker's Jugendtraum is relevant. But maybe that's working too hard. You could try asking this as a separate question.
Jul
2
comment Can we recover homology from cohomology
For the first question the answer is no in general but yes for spaces with levelwise finitely generated homology, which includes all compact manifolds but not all complex manifolds (e.g. $\mathbb{C}$ minus countably many points). For the second question I'm fairly confident the answer is known to be yes but I don't know examples off the top of my head.