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Added to simple group an example I was given on MO: the simple group of cardinality $\kappa$, given by taking the smallest normal subgroup of $Aut(\kappa)$ containing the 3-cycles. This is essentially the ’even’ permutations for an infinite set.
This is nice, because I was trying to think of a simple group (or one with only small normal subgroups) with inaccessible cardinality, and some obvious tricks weren’t working.
Added to simple group the easy-to-prove proposition that a directed colimit of simple groups is also simple, which enters into David’s observation above.
Also mean to add at some point the proof of simplicity of finite alternating groups $A_n$ for $n \geq 5$, but need more time to examine which ones out there I think are ’nice’.
I just learned that Volume 7 of the classification has been published !! I updated classification of finite simple groups to reflect this, since it stated (from ten years ago) that six volumes were available.
As part of this morning’s work, I recorded the result at simple group that every group (finite or infinite) embeds in a simple group.
I tried to look into the matter of skepticism (about the classification theorem), especially with regard to Conway, but the only concrete lead I found suggests that the correct word here is “pessimism”, not “skepticism”. Have a look.
Serre talks about his attitude in this talk. I’ve linked to a part where he makes comments about the classification in response to an audience comment, but IIRC he makes several comments through the body of the talk. To paraphrase, he says something like “Ashbacher says there is a proof, because he sees the various pieces that are in the literature, and essentially how they all connect, but for us, I will not consider that as a proof, if there are pieces you have to arrange (after all many more theorems are like that)”. He goes on to say the the Classification is acceptable, if people state it as a hypothesis to a theorem: the theorem is then true modulo the Classification. Serre goes on to say the situation is much worse for the ATLAS, but has since reversed his view on that, given recent independent verification work.
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