Let X be a set, | X | = n and G be a group with a 2 −−transitive action on X. what can be said about the size of G?

Question

Let   X be a set,   |  X  |  =  n and   G be a group with a   2  −−transitive action on   X. what can be said about the size of   G?

Samantha Reid · Accepted Answer

Since   G is   2  −transitive, its order is divisible by   n  ×  (  n  −  1  ), not just bounded below by, and so it would be interesting to bound                           |        G              |                    (        n        ×        (        n        −        1        )        )              .If n is not a prime power, then it is quite possible for the lower bound to be huge. The smallest reasonable non-prime,   n  =  6  , has its smallest   2  −transitive group with order   60  =  6  ×  5  ×  2. The next,   n  =  10  ,, has its smallest   2  −transitive group with order   10  ×  9  ×  4. For most   n, the smallest multiple is                     (        n        −        2        )        !            2        , that is, the alternating group on   n points is the smallest   2  −transitive group. This happens already at   n  =  22  ,  33  ,  34  ,  35  , and asymptotically takes over.So on the one hand the lower bound for a   2  −transitive group on n points is   n  ×  (  n  −  1  ) for prime powers   n, but for most   n the lower bound is                     n        !            2        .

Tristian Velazquez · Accepted Answer

When   n is a prime (or indeed a prime power), there are two-transitive subgroups of Sym  (  X  ) (where   |  X  |  =  n) of order   n  (  n  −  1  ). In any case   n  (  n  −  1  ) is a lower bound for the order of   G, since this is the number of ordered pairs   (  x  ,      x    ′    ) of distinct elements of   X. Alas I can&#039;t see where your       n    2    +  ncomes from.

Let X be a set, | X | = n and G be a group with a 2 −<!-

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