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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 nonidentity 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 semidihedral 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) BrauerFowler 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 