bio | website | boolesrings.org/nickgill |
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location | San Jose, Costa Rica | |
age | 37 | |
visits | member for | 5 years, 2 months |
seen | Dec 16 at 2:31 | |
stats | profile views | 1,866 |
I'm a visiting professor at the Universidad de Costa Rica.
Dec 16 |
awarded | Nice Answer |
Dec 15 |
reviewed | Approve textbooks on modern algebraic geometry for 21st-century starters |
Dec 15 |
comment |
Why do sporadic simple groups have so few conjugacy classes?
This list is very interesting. It suggests to me almost the opposite of what the OP asked, namely that the number of conjugacy classes in a simple group is a very well-behaved statistic. I wonder whether one could prove an absolute upper bound for your function $f(G)$, as $G$ ranges over all of the (non-alternating) simple groups? This seems too much to ask, but you never know... I presume that $f(G)$ is unbounded for $A_n$? Could one also prove that $f(G)$ takes its minimal value when $G=M_{22}$? |
Dec 15 |
reviewed | Approve Intuitionistic logic as quantization of classical logic? |
Dec 15 |
reviewed | Approve Models of intuitionistic linear logic that reflect the resource interpretation |
Dec 15 |
revised |
Why do sporadic simple groups have so few conjugacy classes?
extra comment |
Dec 15 |
comment |
Why do sporadic simple groups have so few conjugacy classes?
@GeoffRobinson, this is interesting. Indeed, given that this phenomenon is so general, the rate of convergence becomes interesting when considering the "non-abelian-ness" mentioned in the original question. The Liebeck-Pyber paper is very strong in this regard, but of course such a notion makes no sense in connection with the sporadic groups. |
Dec 15 |
revised |
Why do sporadic simple groups have so few conjugacy classes?
caveat about small rank groups inserted |
Dec 15 |
answered | Why do sporadic simple groups have so few conjugacy classes? |
Dec 11 |
comment |
Generating finite simple groups with $2$ elements
Nice answer. One comment: A group $G$ that has the property that for any element $x$ in $G\setminus\{1\}$ there is an element $y$ such that $\langle x,y\rangle=G$ is said to be $\frac32$-generated. The fact that all finite simple groups are $\frac32$-generated was also proved by Stein (in addition to Guralnick & Kantor, as you mention in your answer). The relevant reference is: Stein, Alexander, $1\frac12$-generation of finite simple groups. Beiträge Algebra Geom. 39 (1998), no. 2, 349–358. |
Dec 10 |
revised |
orders of maximal abelian subgroups
added group theory tag |
Dec 1 |
comment |
Conjugacy classes in lie type group
For field automorphisms: the definition of a field automorphism of a Chevalley group is that it is an $Aut(S)$ conjugate of en element of $\Phi_S$ where $\Phi_S$ is the cyclic subgroup of "entry-by-entry automorphisms". So the conjugacy structure is a triviality. For Steinberg groups and the others the definitions are similar (see Def. 2.5.13 of GLS3). |
Dec 1 |
comment |
Conjugacy classes in lie type group
There are loads of references for this sort of thing - for $PSL_n(q)$ you can use rational forms, for the other classical groups there are variations on rational forms due to MacDonald and also to Wall. For groups of Lie type in full generality I would go to Carter's Finite groups of Lie type. The ATLAS is also very helpful for individual small groups. |
Nov 29 |
comment |
Automorphism of simple lie type groups
One last comment: I presume you mean "generated by all inner and diagonal automorphisms of $S$ that lie in $G$", otherwise I'm not sure that the definition of $G_0$ makes sense. |
Nov 28 |
comment |
Automorphism of simple lie type groups
Checking the atlas for $p=5$ implies that $y$ and $xy$ are not conjugate in that case. So there's one counterexample (with Def 2.5.13 of GLS3). |
Nov 28 |
comment |
Automorphism of simple lie type groups
Possible counterexample: consider $S=PSL_2(p^2)$ for some odd prime $p$. Let $G$ be the group $\langle S, h\rangle$ where $h$ is the product of a diagonal automorphism $x$ and a field automorphism $y$ (both of order $2$). Depending on your definitions, this would be a counter-example so long as $y$ and $xy$ are not conjugate in $Aut(S)$. I guess they aren't but would need to check to be sure. |
Nov 28 |
comment |
Automorphism of simple lie type groups
You probably need to define graph, field and graph-field automorphisms explicitly as there is no universal terminology. (For instance Gorenstein, Lyons & Solomon have two definitions of field and graph auts in their volume 3 - see Warning 2.5.2 of that book.) |
Nov 28 |
comment |
Number of isomorphism types of finite groups
@S.Carnahan - good point, thank you. I have edited accordingly. |
Nov 28 |
revised |
Number of isomorphism types of finite groups
added parantheses as per Scott Carnahan's comment. |
Nov 28 |
awarded | Good Answer |