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Regarding part 2 of your question - "In particular, how do we recognize infinite simple groups?" - I think the answer is that it depends on which infinite simple group you're looking at! Some famous examples:-

• Higman's original construction of an infinite simple group starts with a group with no non-trivial finite quotients. (Roughly, you construct one of these by building in a pair of conjugate elements which would have to have different orders in a finite quotient.) You then proceed to take the quotient by a maximal proper normal subgroup. The result can't be finite, because that would be a non-trivial finite quotient! (There was some discussion of this here.)

• Thompson's groups T and V contain elements of infinite order!

• Tarski Monsters are infinite because of Sylow's Theorems. Every proper subgroup is of prime order p, so Sylow's Theorems tell you that if it were finite then it would be cyclic, which it isn't by construction.

Do you have a particular reason to think that your groups are simple? What do you know about the kernel of the map from the Coxeter group?

EDIT: Just wanted to emphasize that of course, of the examples listed, only Thompson's Groups happen to be finitely presented. Finitely presented infinite simple groups are pretty special.

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Regarding part 2 of your question - "In particular, how do we recognize infinite simple groups?" - I think the answer is that it depends on which infinite simple group you're looking at! Some famous examples:-

• Higman's original construction of an infinite simple group starts with a group with no non-trivial finite quotients. (Roughly, you construct one of these by building in a pair of conjugate elements which would have to have different orders in a finite quotient.) You then proceed to take the quotient by a maximal proper normal subgroup. The result can't be finite, because that would be a non-trivial finite quotient! (There was some discussion of this here.)

• Thompson's groups T and V contain elements of infinite order!

• Tarski Monsters are infinite because of Sylow's Theorems. Every proper subgroup is of prime order p, so Sylow's Theorems tell you that if it were finite then it would be cyclic, which it isn't by construction.

Do you have a particular reason to think that your groups are simple? What do you know about the kernel of the map from the Coxeter group?