bio  website  boolesrings.org/nickgill 

location  San Jose, Costa Rica  
age  38  
visits  member for  5 years, 9 months 
seen  2 days ago  
stats  profile views  2,054 
I'm a visiting professor at the Universidad de Costa Rica.
2d

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SO$(4)$ (& SO$(n)$) characterization?
These notes assert that you can find a list of the finite subgroups of $SO(4)$ in Homographies, Quaternions and Rotations by P. Du Val. I don't have a copy of this reference, so can't confirm it. It would be nice to have an explicit list as an answer to the OP's question! 
Jul 29 
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Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?
This result (about minimal FSG's) is a special case of Theorem B of A NEW SOLVABILITY CRITERION FOR FINITE GROUPS by Dolfi, Guralnick, Herzog and Praeger. They also state a conjecture in the final section which pertains directly to the OP's original question. It's slightly too long for me to write down in a comment, but you can find it here  arxiv.org/pdf/1105.0475.pdf 
Jul 28 
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Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?
You should look at Claude Marion's lovely paper On triangle generation of finite groups of Lie type which studies triangle groups $(a,b,c)$ where $a,b$ and $c$ are prime. You may well be able to extract what you need from that paper. 
Jul 27 
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Isotropic subspaces in a symplectic vectorspace over $GF(q)$
You only need to prove the statement for $r=n$, as the rest will follow directly... And the statement is true when $2n=4$ and the field is ${\mathbb F}_3$  see p.33 of this: math.lsu.edu/~hoffman/papers/spreads4.pdf (Whether that provides any evidence for the statement in general, I couldn't say. The group ${\rm Sp}_4(3)$ is a bit special.) 
Jun 17 
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Transitivity on $\mathbb{N}_0$ — a 42 problem
That statement looks crucial to me! If you can write down a proof of that, then this would be a most illuminating addition to the question. To me, the number "42", despite Douglas Adam's assertions to the contrary, seems a rather random element in the setup that you have in this question. Your comment changes all that, though (for me). 
Jun 17 
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minimal polynomial of unipotents in orthogonal group
BTW, I've voted for this question to be reopened. In its current form it seems fine to me. 
Jun 17 
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minimal polynomial of unipotents in orthogonal group
You'd be better of using the matrix with $1$'s on the antidiagonal. Then you can take your unipotents to be uppertriangular, and this question should yield to direct calculation  my first thought is that the minimal polynomial of a unipotent $g$ will just depend on the dimension of the largest Levi subgroup that contains $g$ 
Jun 17 
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Transitivity on $\mathbb{N}_0$ — a 42 problem
Thanks for your replies Stefan. One last thing: can you prove your conjecture for the situation where no generator interchanges residue classes with different moduli? In light of your last comment, I guess you want to show that no such group can have an orbit containing {0,...,42}. This would already seem kind of surprising to me. (Although maybe that's just my ignorance...) 
Jun 16 
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Transitivity on $\mathbb{N}_0$ — a 42 problem
Sorry, Stefan, my last question was not clear: I wanted to know if you can easily tell when $G=\langle a,b,c\rangle$ is infinite, for arbitrary choices of $a,b$ and $c$? 
Jun 16 
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Transitivity on $\mathbb{N}_0$ — a 42 problem
Do you have more information about the presentation of such a group? For instance, what is the order of the product of two of these involutions? Looking at the group $G$ in your counterexample for $n=41$, I'm seeing 2,3 and 7 and thinking Hurwitz group. I don't know if there really is a connection, though... If there were, then perhaps the theory of triangle groups might help... Do you even know that, under your supposition, the group $G$ is infinite? 
Jun 10 
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Is there a big solvable subgroup in every finite group?
By taking quotients, one can assume that $F(G)$ is trivial. Thus $F^*(G)$ is a central product of a bunch of quasisimples. If that helps. 
Jun 10 
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Is there a big solvable subgroup in every finite group?
Pedantry: I think the A_8 example should be $(S_4\wr S_2)\cap A_8$ rather than $(S_4\cap S_4)\cap A_8$. (The former is not maximal in $A_8$, although it may be big.) 
May 16 
awarded  Nice Answer 
May 5 
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Extensions of $SL(2,\mathbb{F}_q)$
The outer atuomorphism group of $SL_2(2^n)$ is certainly the Galois group, as you say. One source is Kleidman & Liebeck's opening couple of chapters. (Dieudonne also deals with this I believe and perhaps he is responsible for the result originally??) I have a ecopy of K&L should you need it. 
May 5 
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Extensions of $SL(2,\mathbb{F}_q)$
If $q$ is even and bigger than $2$, then $SL(2,q)$ is simple and things are easy: you always have $SL(2,q)\times C_2$, and if $q$ is a square you will have an almost simple group, and these are all the possibilities. If $q$ is odd, then you are talking about bicyclic extensions of $PSL(2,q)$ and the notion of ISOCLINISM makes things a little more tricky. I recommend you read the introduction of the ATLAS for an excellent discussion of this. (I have an ecopy of the ATLAS if you need it.) 
Apr 24 
revised 
Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?
Corrected some spelling and English. 
Apr 23 
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learning DeligneLusztig theory
I found it useful to read Green's original paper on the characters of $GL_n(q)$. It isn't DeligneLusztig theory as such but it prefigures it. (And, from what I recall, doesn't need too much background.) 
Apr 23 
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learning DeligneLusztig theory
In the first section on Character Theory of Finite Groups I'd also recommend Marty Isaacs' book. (Indeed, a general mathematical rule of thumb: always read Marty Isaacs' books.) 
Apr 23 
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learning DeligneLusztig theory
That is a brilliant answer. 
Apr 9 
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When few simple conditions yield a unique intricate structure
How do you describe the sporadic simple groups by a few easy conditions? 