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# Transitive Group

Transitivity is a result of the symmetry in the group. A group is called transitive if its group action (understood to be a subgroup of a permutation group on a set ) is transitive. In other words, if the group orbit is equal to the entire set for some element , then is transitive.

A group is called k-transitive if there exists a set of elements on which the group acts faithfully and -transitively. It should be noted that transitivity computed from a particular permutation representation may not be the (maximal) transitivity of the abstract group. For example, the Higman-Sims group has both a 2-transitive representation of degree 176, and a 1-transitive representation of degree 100. Note also that while -transitivity of groups is related to -transitivity of graphs, they are not identical concepts.

The symmetric group is -transitive and the alternating group is -transitive. However, multiply transitive finite groups are rare. In fact, they have been completely determined using the classification theorem of finite groups. Except for some sporadic examples, the multiply transitive groups fall into infinite families. Certain subgroups of the affine group on a finite vector space, including the affine group itself, are 2-transitive. Some of these are summarized below.

The multiply transitive groups fall into six infinite families, and four classes of sporadic groups. In the following enumeration, is a power of a prime number.

1. Certain subgroups of the affine group on a finite vector space, including the affine group itself, are 2-transitive.

2. The projective special linear groups are 2-transitive except for the special cases with even, which are actually 3-transitive.

3. The symplectic groups defined over the field of two elements have two distinct actions which are 2-transitive.

4. The field of elements has an involution , so , which allows a Hermitian form to be defined on a vector space on . The unitary group on , denoted , preserves the isotropic vectors in . The action of the projective special unitary group is 2-transitive on the isotropic vectors.

5. The Suzuki group of Lie type is the automorphism group of a Steiner system, an inversive plane of order , and its action is 2-transitive.

6. The Ree group of Lie type is the automorphism group of a Steiner system, a unital of order , and its action is 2-transitive.

7. The Mathieu groups and are the only 5-transitive groups besides and . The groups and are 4-transitive, and is 3-transitive.

8. The projective special linear group has another 2-transitive action related to the Witt geometry .

9. The Higman-Sims group HS is 2-transitive.

10. The Conway group is 2-transitive.

Other 3-transitive groups include acting on 8 items, as generated by the permutations , , and ; and acting on 12 items, as generated by the permutations , , and .

Arc-Transitive Graph, Edge-Transitive Graph, k-Transitive Group, Leech Lattice, Mathieu Groups, Simple Group, Steiner System, Transitive Group Action, Vertex-Transitive Graph

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Nick Wedd

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## References

Dixon, J. and Mortimer, B. Permutation Groups. New York: Springer-Verlag, 1996.Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, p. 27, 1993.Klüners, J. and Malle, G. "Explicit Galois Representation of Transitive Groups of Degree Up to 15." J. Symb. Comput. 30, 675-716, 2000.

Transitive Group

## Cite this as:

Rowland, Todd; Wedd, Nick; and Weisstein, Eric W. "Transitive Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TransitiveGroup.html