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1d
comment Determining if a matrix is orthogonal
For the first part: Since you need $\beta X\beta^{-1} = (X^{t})^{-1}$, it is certainly necessary that $X$ and $X^{-1}$ have the same characteristic polynomial.
2d
comment What conditions imply that a function over $\mathbb{Z}$ is a polynomial?
If you have the function value at enough points, and it really is polynomial, then the polynomial given by Lagrange interpolation will eventually stabilize to the right polynomial, and adding extra points and function values will not change it. If that doesn't happen, the function isn't polynomial.
2d
comment What conditions imply that a function over $\mathbb{Z}$ is a polynomial?
Lagrange interpolation if you know enough values of your function. You implicitly did this sort of thing with your sin example. A polynomial function which takes the value 0 infinitely often is identically 0, which sin(x) is not.
2d
comment What conditions imply that a function over $\mathbb{Z}$ is a polynomial?
A polynomial of degree $n$ is uniquely determined from its values at $n+1$ distinct points.
Jun
26
revised Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
minor textual change to avoid repetition
Jun
25
revised Classification of groups in which the centralizer of every non-identity element is cyclic
typo
Jun
23
revised Invariants of Symmetric group
minor typos
Jun
23
comment On Thompson conjecture
@DerekHolt : Yes, I think that is what is intended. As you can see, I was caught out too.
Jun
23
comment On Thompson conjecture
@DerekHolt : That is true within $A_{n}$, but is it clear that that property carries over to $G$?
Jun
23
comment On Thompson conjecture
I was mistaken.
Jun
23
comment Next steps on formal proof of classification of finite simple groups
I do not know what is being pursued at present in this area with regard to formal proofs. Long and difficult proofs which I think might be viable candidates, and which are somehow outside the "generic" part of the CFSG proof are the (separate) classification of groups with dihedral and semi-dihedral Sylow $2$-subgroups (low rank Sylow $2$-subgroups in general).
Jun
23
comment Next steps on formal proof of classification of finite simple groups
The proof of the (fundamental, I agree) Brauer-Fowler Theorem is quite short, and not especially difficult- the key to that result is a great insight. Given that initial insight, the proof can be followed without too much difficulty by specialists.
Jun
20
revised Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup?
Added section about application of answer.
Jun
20
revised Does a finite simple group of order divisible by $60$ have $A_{5}$ as a subgroup?
added 4 characters in body
Jun
19
awarded  Necromancer
Jun
17
revised Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
typo
Jun
17
revised Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
explained char 0 case, clarified
Jun
17
revised Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
Illustrated sufficiency in finite characteristic
Jun
17
comment Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
I found it a bit unclear exactly what you were asking for. To be more precise than my answer, you certainly need the derived group $[G,G]$ to be a $p$-group when $k$ has characteristic $p >0.$ That may also be sufficient for $k$ large enough, I might expand my answer.
Jun
17
revised Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?
gave another example