# Doubly-transitive groups

I want to know what all doubly-transitive groups look like. Do you know some good reference where I can read about it?

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Solvable groups are classical and non-solvable groups were classified by Hering (using the classification of finite simple, which is guess is OK by now). See en.wikipedia.org/wiki/2-transitive_group for reference. – Torsten Ekedahl Jul 18 '10 at 12:41
I also want a proof for this classification. Do you know if there exist some simple source explaining it? – Klim Efremenko Jul 18 '10 at 13:06
I believe Robinson's Group Theory book does the solvable case and, modulo the CFSG, Cameron's "Permutation Groups" does the non-solvable case. I can't check either of these right now, though, so I could be remembering wrong. – Steve D Jul 18 '10 at 13:45
This kind of classification isn't in principle neat or simple; eventually it also depends on knowing what the finite simple groups are. Probably the most useful recent textbook source is the 1999 LMS Student Text (Cambridge) Permutation Groups by Peter J. Cameron. – Jim Humphreys Jul 18 '10 at 13:53
The answers given so far indicate that your "good reference" and "all" may be elusive. – Jim Humphreys Jul 18 '10 at 22:12

## 2 Answers

In Section 7.7 "The Finite 2-transitive Groups" of the book Permutation groups by John D. Dixon and Brian Mortimer, the authors describe the complete list of finite 2-transitive groups without proofs but with references.

They list eight infinite families: the alternating, symmetric, affine and projective groups in their natural actions, as well as the less known groups of Lie type: the symplectic groups, the Suzuki groups, the unitary groups and the Ree groups. The symplectic groups have two distinct 2-transitive actions, the last three classes are 2-transitive on the sets of points in their action on appropriate Steiner systems. Additional there are 10 sporadic examples of 2-transitive groups.

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The solvable case was proved by Huppert in 1957. It is presented in textbook form in chapter XII.7 of Huppert–Blackburn's Finite Groups Vol. 3. This includes full proof that only certain exceptional cases can occur, but not full coverage of the exceptional cases. It includes an outline of Hering (et al.)'s proof of the insoluble case, which is spread over several papers (which are the best source of the proof).

The solvable case is not classical and is not presented in Robinson's textbook. Presumably people are thinking of solvable groups of prime degree. Cameron's book does not contain any proof, and its tables are partial so be careful of relying on it for details. Obviously the classical textbooks can only present partial pictures as the results were not known at the time.

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