Multiply transitive group action

A group ${\displaystyle G}$acts 2-transitively on a set ${\displaystyle S}$ if it acts transitively on the set of distinct ordered pairs ${\displaystyle \{(x,y)\in S\times S:x\neq y\}}$. That is, assuming (without a real loss of generality) that ${\displaystyle G}$ acts on the left of ${\displaystyle S}$, for each pair of pairs ${\displaystyle (x,y),(w,z)\in S\times S}$ with ${\displaystyle x\neq y}$ and ${\displaystyle w\neq z}$, there exists a ${\displaystyle g\in G}$ such that ${\displaystyle g(x,y)=(w,z)}$.

The group action is sharply 2-transitive if such ${\displaystyle g\in G}$ 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, ${\displaystyle gx=w}$ and ${\displaystyle gy=z}$, since the induced action on the distinct set of pairs is ${\displaystyle g(x,y)=(gx,gy)}$.

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.

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

Let ${\displaystyle S_{n}}$ be the symmetric group acting on ${\displaystyle \{1,...,n\}}$, then the action is sharply n-transitive.

The group of n-dimensional similarities acts 2-transitively on ${\displaystyle \mathbb {R} ^{n}}$. In the case ${\displaystyle n=1}$ this action is sharply 2-transitive, but for ${\displaystyle n>1}$ it is not.

The group of n-dimensional projective transformsalmost acts sharply (n+2)-transitively on the n-dimensional real projective space${\displaystyle \mathbb {RP} ^{n}}$. 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 ${\displaystyle \mathbb {RP} ^{n}}$.

Todo grupo 2-transitivo es un grupo primitivo , pero no a la inversa. Todo grupo de Zassenhaus es 2-transitivo, pero no a la inversa. Los grupos 2-transitivos resolubles fueron clasificados por Bertram Huppert y se describen en la lista de grupos lineales finitos transitivos . Los grupos insolubles fueron clasificados por Hering ( 1985 ) utilizando la clasificación de grupos simples finitos y son todos grupos casi simples .

• Multiplicar grupo transitivo

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