6,349 reputation
1445
bio website boolesrings.org/nickgill
location San Jose, Costa Rica
age 38
visits member for 5 years, 7 months
seen yesterday

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


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?
Mar
18
comment Maximal abelian subgroup of general linear groups
This might be helpful: mathnet.ru/php/…
Mar
11
comment Experimenting with the spider relator
This paper - ftp.mathe2.uni-bayreuth.de/axel/papers/ivanov:the_monster.ps.gz - seems to assert that ${Y222}\cong 3^5. O_5(3)$. By the way, your questions would be much easier to read if you used LaTeX!
Mar
6
comment Matrices congruent to each other via a permutation
... Actually, it's possible that you can then just consider super-diagonal entries and order these to be strictly increasing. You'd need to prove that this is (a) possible, and (b) unique (neither of which I'm sure about). Supposing this works, then iterating (considering super-super-diagonals etc) would yield a canonical form.
Mar
6
comment Matrices congruent to each other via a permutation
A first thought: Consider an integer matrix whose diagonal entries are all distinct. You can use permutation matrices to order these entries so that they are strictly increasing as you go down the diagonal. There will only be one such matrix in the equivalence class that you describe so you could consider this a canonical form for this type of matrix. (Note that this works fine for real matrices, not just integer matrices.) I'm not sure how to extend this to matrices with repeated diagonal entries...
Feb
26
comment How do small central extensions drop the dimension of a faithful representation?
Ben, you might be interested in this paper - arxiv.org/pdf/1408.1649.pdf - by Britnell, Saunders and Skyner. It looks at a similar phenomenon for $p$-groups - when the quotient of a $p$-group has minimal permutation representation larger than the original.
Feb
12
revised Can any finite distributive weighted lattice be realized by inclusion of groups?
typos fixed
Feb
11
answered Can any finite distributive weighted lattice be realized by inclusion of groups?
Feb
11
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
Well that sounds promising. My thinking is that the three element lattice is equivalent to having an imprimitive permutation group on a set of size $mn$ with a unique system of imprimitivity. There are no doubt many of these, but the wreath product is certainly one such. And I can't see a reason why iterating wouldn't work, so that probably sorts chains out. I'll try and write a proper answer later today when I have time.
Feb
10
reviewed Approve Unable to find any information regarding this fact (Frey, elliptic curves)
Feb
10
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
I could be wrong, but if you let $G=S_m \wr S_n$ and consider the natural action on $mn$ points, then the point-stabilizer has index $mn$ and there is a unique subgroup in between of index $n$. Sorry, no time to check this through but if it works it deals with $(mn,n,1)$.... And iterated wreath products will probably deal with chain lattices...
Feb
10
comment Can any finite distributive weighted lattice be realized by inclusion of groups?
Have you tried $(36,6,1)$? That seems like the first tricky $(n^2,n,1)$ case...
Feb
6
comment Simple Hurwitz Groups of order less than 10^7
Thomas, note that $B_2(q)$ and $C_2(q)$ are isomorphic.