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bio website boolesrings.org/nickgill
location San Jose, Costa Rica
age 38
visits member for 5 years, 9 months
seen 2 days ago

I'm a visiting professor at the Universidad de Costa Rica.


2d
comment 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
comment 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
comment 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
comment 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
comment 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 set-up that you have in this question. Your comment changes all that, though (for me).
Jun
17
comment 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
comment minimal polynomial of unipotents in orthogonal group
You'd be better of using the matrix with $1$'s on the anti-diagonal. Then you can take your unipotents to be upper-triangular, 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
comment 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
comment 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
comment 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 counter-example 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
comment 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 quasi-simples. If that helps.
Jun
10
comment 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
comment 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 e-copy of K&L should you need it.
May
5
comment 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 e-copy 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
comment learning Deligne-Lusztig theory
I found it useful to read Green's original paper on the characters of $GL_n(q)$. It isn't Deligne-Lusztig theory as such but it prefigures it. (And, from what I recall, doesn't need too much background.)
Apr
23
comment learning Deligne-Lusztig 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
comment learning Deligne-Lusztig theory
That is a brilliant answer.
Apr
9
comment When few simple conditions yield a unique intricate structure
How do you describe the sporadic simple groups by a few easy conditions?