Mathieu Group - Multiply Transitive Groups

Multiply Transitive Groups

Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number k, a permutation group G acting on n points is k-transitive if, given two sets of points a₁, ... a_k and b₁, ... b_k with the property that all the a_i are distinct and all the b_i are distinct, there is a group element g in G which maps a_i to b_i for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive).

M₂₄ is 5-transitive, and M₁₂ is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of m points, and accordingly of lower transitivity (M₂₃ is 4-transitive, etc.).

The only 4-transitive groups are the symmetric groups S_k for k at least 4, the alternating groups A_k for k at least 6, and the Mathieu groups M₂₄, M₂₃, M₁₂ and M₁₁. (Cameron 1999, p. 110) The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.

It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k + 2 respectively), and M₁₂ and M₁₁ are the only sharply k-transitive permutation groups for k at least 4.

Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups. The Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,F_q), which is sharply 3-transitive (see cross ratio) on elements.