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 group
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 nonzero finite vector
space, including the affine group itself, are
2-transitive. Some of these are summarized below.
The following are examples of multiply transitive groups from infinite families and sporadic groups. Here
is a power of a prime
number, subject to the restrictions for each family.
1. Certain subgroups of the affine group on a nonzero finite vector space, including the affine group itself, are 2-transitive.
2. The projective special linear groups 
with 
are 2-transitive except for the special cases 
with 
even, which are actually 3-transitive.
3. The symplectic groups 
with 
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 one-dimensional isotropic subspaces in
. The action
of the projective special unitary group 
is 2-transitive on the one-dimensional
isotropic subspaces.
5. The Suzuki group of Lie type 
, for 
with 
, is a subgroup of the automorphism group of an 
Steiner system,
an inversive plane of order 
, and its action is 2-transitive.
6. The Ree group of Lie type 
, for 
with 
, is a subgroup of the automorphism group of an 
Steiner system,
a unital of order 
, and its action is 2-transitive.
7. The Mathieu groups 
and 
are the only 5-transitive finite
groups apart from the natural actions of the
symmetric groups 
for 
and alternating groups 
for 
. 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 
.
