Multiply transitive group action explained

A group

acts 2-transitively on a set

if it acts transitively on the set of distinct ordered pairs

\{(x,y)\inS x S:x ≠ y\}

. That is, assuming (without a real loss of generality) that

acts on the left of

, for each pair of pairs

(x,y),(w,z)\inS x S

with

x ≠ y

and

w ≠ z

, there exists a

g\inG

such that

g(x,y)=(w,z)

The group action is sharply 2-transitive if such

g\inG

is unique.

A 2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply 2-transitive group.

Equivalently,

gx=w

and

gy=z

, since the induced action on the distinct set of pairs is

g(x,y)=(gx,gy)

The definition works in general with k replacing 2. Such multiply transitive permutation groups can be defined for any natural number k. Specifically, 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. The Mathieu groups are important examples.

Examples

Every group is trivially 1-transitive, by its action on itself by left-multiplication.

Let

S_n

be the symmetric group acting on

\{1,...,n\}

, then the action is sharply n-transitive.

The group of n-dimensional homothety-translations acts 2-transitively on

\Rⁿ

RPⁿ

. The almost is because the (n+2) points must be in general linear position. In other words, the n-dimensional projective transforms act transitively on the space of projective frames of

RPⁿ

Classifications of 2-transitive groups

Every 2-transitive group is a primitive group, but not conversely. Every Zassenhaus group is 2-transitive, but not conversely. The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups. The insoluble groups were classified by using the classification of finite simple groups and are all almost simple groups.

Multiply transitive group action explained

Examples

Classifications of 2-transitive groups

See also