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The classification of finite simple groups is known to be very very long. But I was wondering: is there somehow a classification of the infinite simple groups, or at least a beginning of a classification?

Here are a few examples. Such a classification might appear far more difficult than the one for finite simple groups, but sometimes the infinite case is easier, as shown there. For instance, the classification of algebraically closed fields of cardinality $2^{\aleph_0}$ is just based on the characteristic, if I'm not mistaken.

Any comment is welcome. Thank you!

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Alphonse
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  • ZOMG I'm famous ! – Asinomás Aug 17 '16 at 21:01
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    @Alphonse: Just a remark. There is a total classification of algebraically closed fields. They depend (up to isomorphism) only on transcendence degree and characteristic. – Kyle Gannon Aug 17 '16 at 21:08
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    As far as I can tell, a classification of any kind is hopeless even for simple finitely generated groups. – Moishe Kohan Aug 18 '16 at 01:49
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    I agree with @studiosus, but you might be interested in this answer which discusses a program to attempt to classify certain kinds of simple groups with some sort of finiteness condition. I don't know if that program is hopeless –  Aug 18 '16 at 03:55
  • @PaulPlummer: Yes, I have heard about this program. I have no idea how realistic it is (I know very little about logic) but given the list of people working on it and progress that they have made, I would never consider it as hopeless. The class of groups of finite Morley rank is essentially disjoint from finitely generated groups (apart from finite groups). – Moishe Kohan Aug 18 '16 at 05:30

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