bio | website | boolesrings.org/nickgill |
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location | San Jose, Costa Rica | |
age | 37 | |
visits | member for | 5 years |
seen | 9 hours ago | |
stats | profile views | 1,779 |
I'm a visiting professor at the Universidad de Costa Rica.
Oct 20 |
awarded | Yearling |
Oct 10 |
comment |
Equivalence classes of (2,3)-pairs in PSL(2,q)
My mistake! Alexander's answer below shows my error - the triples fuse in $S_5$ (explaining Robin's calculation in the question) but, of course, there is no $S_5$ in $PGL_2(q)$... |
Oct 2 |
comment |
Equivalence classes of (2,3)-pairs in PSL(2,q)
One further thing: For products of order equal to $2$, $3$ and $4$, I guess the problem reduces to studying the number of orbits in $Aut(G)$ of subgroups isomorphic to $S_3$, $A_4$ and $S_4$. I'm guessing there's only one such subgroup in the latter two cases, not sure about more generally. (This should be a tractable problem though.) For product $6$ the pair commutes so this is easy. For product $7$ one is studying Hurwitz groups and so the pair will generate $PSL_2(p)$ where $p$ is the prime. Now one can check how many non-conjugate triples generate any given $PSL_2(p)$. |
Oct 2 |
comment |
Equivalence classes of (2,3)-pairs in PSL(2,q)
.. Thus I think the two orbits found by Stefan will fuse in $PGL_2(11)$. And, in general, there will be one orbit of $(2,3)$-pairs with product equal to $5$ in the automorphism group of $G$. For other product orders things will be different I guess. |
Oct 2 |
comment |
Equivalence classes of (2,3)-pairs in PSL(2,q)
I'm slightly confused - I think this question neglects the OP's consideration of the automorphism group of $G$. In the case where you have a $(2,3)$-pair whose product is $5$ they will generate a group isomorphic to $A_5$. Subgroups of this form exist in $PSL_2(q)$ exactly when $q=-1,0,1\pmod 5$. What is more if one considers the automorphism group of $G$, then all subgroups isomorphic to $A_5$ are conjugate. Now Robin's calculations for $q=5$ suggest that all such pairs should be conjugate in the automorphism group of $PSL_2(q)$. |
Sep 30 |
awarded | Explainer |
Sep 24 |
comment |
A generalization of real characters on a group
+1 for the first sentence. |
Sep 17 |
comment |
Normal subgroup of classical groups
@M.B Sorry, no idea. I don't know anything about the fields you describe... |
Sep 10 |
comment |
Ends of Coxeter Groups
@YCor, of course your comment is entirely reasonable. There are plenty more results in the cited text that give more explicit information (see especially Thm. 8.7.3)... but I don't want to write them all out! |
Sep 10 |
answered | Ends of Coxeter Groups |
Sep 10 |
comment |
Ends of Coxeter Groups
You can find some information on ends of Coxeter groups in this paper by Mihalik: sciencedirect.com/science/article/pii/0022404995001174 |
Sep 5 |
comment |
existence of a finite group which is the union of self normalizing subgroups
That's a cool fact about Carter subgroups. I didn't know about those. |
Sep 5 |
awarded | Nice Answer |
Sep 4 |
comment |
(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples
It might help to give a little motivation also - why is this particular triple of elements of interest?? |
Sep 4 |
comment |
(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples
what do you mean "there are elements of order $n$ in $G$, but not in any of its proper subgroups"? This will only be true when $G$ is cyclic of order $n$, but I guess you're still assuming $G=PSL(2,q)$... Perhaps you mean subspace subgroups?? |
Sep 4 |
reviewed | Approve suggested edit on Trace inequality for matrices with determinant 1 |
Sep 4 |
revised |
Which finite simple groups can be characterized by their action on a small set?
Added a discussion of the general setting. |
Sep 4 |
answered | existence of a finite group which is the union of self normalizing subgroups |
Sep 2 |
answered | Which finite simple groups can be characterized by their action on a small set? |
Sep 2 |
comment |
Which finite simple groups can be characterized by their action on a small set?
... By the way I have an e-copy of the LPS-memoir - email me if you want it.... |