Abstract:
The main aim of this research is to determine the ranks, subdegrees, and the
suborbital graphs of the symmetric group Sn acting on unordered r-element subsets
of X = {1, 2, 3, ..., n}. These areas have not received much attention, in fact
most of the research has been focused on the action of Sn on unordered pairs. In
1970, Higman proved that Sn, n ≥ 4 acts as a rank 3 group on X(2), with subdegrees
1, 2(n − 2),
0
B@
n − 2
2
1
CA
. In this study, it is shown that Sn acts transitively
and primitively on X(r)(r-element subsets of X). The ranks and suborbits of Sn
acting on X(4) and X(5) are determined, after which it is proved that the rank of
Sn acting on X(r) is r + 1 if n ≥ 2r. The suborbits of Sn acting on X(r) are all
self paired is shown. It is also proved that the subdegrees of Sn acting on X(r) are
1, r
0
B@
n − r
r − 1
1
CA
,
0
B@
r
2
1
CA
0
B@
n − r
r − 2
1
CA
,
0
B@
r
3
1
CA
0
B@
n − r
r − 3
1
CA
, ...,
0
B@
r
r − 1
1
CA
0
B@n
−
r
1
1
CA
,
0
B@
n − r
r
1
C A
,
after which the subdegrees are arranged in an ascending order. The suborbital
graphs corresponding to the suborbits of Sn are then constructed and their properties
analysed . It is shown that when Sn acts on X(r), its suborbital graphs are
undirected and have girth three if n ≥ 3r.
