Let
X
X
be a finite set such that
|
X
|
=
n
|X|=n
and let
i
⩽
j
⩽
n
i\leqslant j \leqslant n
. A group
G
⩽
S
n
G\leqslant \mathcal {S}_{n}
is said to be
(
i
,
j
)
(i,j)
-homogeneous if for every
I
,
J
⊆
X
I,J\subseteq X
, such that
|
I
|
=
i
|I|=i
and
|
J
|
=
j
|J|=j
, there exists
g
∈
G
g\in G
such that
I
g
⊆
J
Ig\subseteq J
. (Clearly
(
i
,
i
)
(i,i)
-homogeneity is
i
i
-homogeneity in the usual sense.)
A group
G
⩽
S
n
G\leqslant \mathcal {S}_{n}
is said to have the
k
k
-universal transversal property if given any set
I
⊆
X
I\subseteq X
(with
|
I
|
=
k
|I|=k
) and any partition
P
P
of
X
X
into
k
k
blocks, there exists
g
∈
G
g\in G
such that
I
g
Ig
is a section for
P
P
. (That is, the orbit of each
k
k
-subset of
X
X
contains a section for each
k
k
-partition of
X
X
.)
In this paper we classify the groups with the
k
k
-universal transversal property (with the exception of two classes of
2
2
-homogeneous groups) and the
(
k
−
1
,
k
)
(k-1,k)
-homogeneous groups (for
2
>
k
⩽
⌊
n
+
1
2
⌋
2>k\leqslant \lfloor \frac {n+1}{2}\rfloor
). As a corollary of the classification we prove that a
(
k
−
1
,
k
)
(k-1,k)
-homogeneous group is also
(
k
−
2
,
k
−
1
)
(k-2,k-1)
-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the
k
k
-universal transversal property have the
(
k
−
1
)
(k-1)
-universal transversal property.
A corollary of all the previous results is a classification of the groups that together with any rank
k
k
transformation on
X
X
generate a regular semigroup (for
1
⩽
k
⩽
⌊
n
+
1
2
⌋
1\leqslant k\leqslant \lfloor \frac {n+1}{2}\rfloor
).
The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.