A transitive group, also called a transitive permutation group, is a permutation group 
whose group action on a set 
has a single group
orbit. Equivalently, for every 
, some 
satisfies 
. Transitivity is therefore a property of a particular
permutation action, not only of the abstract group.
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 
.
