Nick Gill
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 Apr 21 comment Regular elements in the torus of a group of Lie type Great, thanks for this, very helpful. Apr 21 comment Regular elements in the torus of a group of Lie type Hi Jay, Thanks for you answer. I'll have to think some more to understand what you've written. I had a quick look at Carter's Proposition 3.6.1 (in Finite Groups of Lie type), and it seems to me that maximally split maximal tori may be non-degenerate for $q=3$ too (at least for some simple groups $G$ and some endomorphisms $F$). Does this seem reasonable? Apr 20 awarded Nice Question Apr 20 comment Finite groups generated by 3 involutions interchanging disjoint residue classes of the integers Do you know if, in cases 7,8 and 9, there is only one group turning up with that maximal order? Any information about, say, solvability of the group(s) in question? Mar 11 comment To determine if a 2 variable symmetric function is addition formula of one variable function or not? @AlexandreEremenko, perhaps you are referring to this paper: ams.org/journals/tran/1927-029-02/S0002-9947-1927-1501393-4/… Mar 10 comment On the structure of groups according to their conjugacy classes You might need to add some motivation for your question. As it stands, it seems rather arbitrary. Mar 9 awarded Nice Answer Mar 9 answered Simplicity of $A_n.$ Mar 9 comment Simplicity of $A_n.$ This document has FIVE proofs of the simplicity of $A_n$, for $n\geq 5$: math.uconn.edu/~kconrad/blurbs/grouptheory/Ansimple.pdf Mar 9 comment Simplicity of $A_n.$ ... Iwasawa's Criterion will also do all the other alternating groups if you consider the action on 3-subsets. (Actually, rather than say all the classicals, perhaps I should hedge my bets and exclude the orthogonals -- I can't immediately remember the way the proof goes there.) I wrote a bunch of notes on this by the way that I am happy to send you (although it's all very classical). Mar 9 comment Simplicity of $A_n.$ Using Iwasawa's Criterion would be a good thing to do if you're thinking of generalizing to other FSG's (it's good for all the classicals). Feb 25 comment Applications of the Cayley-Hamilton theorem @RobertIsrael, thanks for your comment.... I had never thought through the practicalities of the various methods... Feb 25 comment Applications of the Cayley-Hamilton theorem I always get my students to use CH to write down a formula for $A^{-1}$, in terms of $I, A, A^2$ etc. I then tell them that this is very useful in practice because it allows you to compute an inverse in a fashion that is not computationally intensive. I must admit, though, that I don't know if this is used in the real world... But, if so, that might be an important application, no?! Feb 11 comment What do named “tricks” share? Another possible entry in the list : the Frattini argument. I thought of it when I read Qiaochu's answer (which I thought was spot on) -- it's a very handy little, um, thingy to know when working with finite groups. Feb 11 comment What do named “tricks” share? Just to add to your list -- one of the commenters above has a trick named after him :-) Nikolov and Pyber used a result of Gowers about solving equations in groups to prove a nice fact about the multiplication of (large) sets in groups. The method they used is known as the Gowers trick. I believe it was Pyber who came up with the name -- you can see the name in use in the literature in several places, for instance in Breuillard's Introduction to approximate groups. Feb 10 comment Sets of matrices which are irreducible but not strongly irreducible @BenjaminSteinberg, could you expand on your comment a little -- I don't quite understand it. Nice question by the way, Ian. Feb 4 awarded Nice Answer Jan 14 comment long root elements fixed by an automorphism in simple lie type group Ah, sorry, I misread your question -- you just want to fix elements in a specific long root subgroup. Well, this will just require that you calculate the centralizer (in ${\rm Aut}(G)$) of all the elements in that group, and take the intersection. This seems pretty straightforward. Jan 14 comment long root elements fixed by an automorphism in simple lie type group I guess you're wanting to fix some split maximal torus before you start defining root subgroups. After that do you want your automorphism to centralize every long root subgroup, or to normalize every long root subgroup? The former certainly can't be done in the case where there is only one length root (e.g ${\rm PSL_n(q)}$), as the long root subgroups generate the simple group. In the other cases (normalizing and/or more than one root length), I'd have to think a little... Jan 6 answered embedding of $O_4^-(q)$ in $U_4(q)$