## Subgroups, Lattice Structures, and the Number of Sylow -Subgroups for Symmetric Groups P

##### Abstract

Subgroups and supergroups of various symmetric groups have been researched on
extensively. Various suggestions by researchers have been provided on how to find
the numbers of Sylow -subgroups. Casadio (1990) has provided the proof for the
third Sylow theorem, which will greatly contribute to the finding of the possible
variety of numbers of the Sylow -subgroups. He stated that the number of the
Sylow -subgroups of a group is congruent to 1 modulo and divides
i.e.
and is gotten by dividing the order of the group with the order of
the Sylow -subgroup. The research will be made up of five chapters. It has
emphasized the number of subgroups, supergroups, lattice structures and the
ascending chains of various symmetric groups. We shall largely study the Sylow -
subgroups in symmetric groups , n =1,2,3,4,5,6,7,8,9,10 which will lead to a
generalization of the number of Sylow -subgroups in any symmetric group and
hence coming up with a formula for getting the number of the Sylow -subgroups of
symmetric groups of any given order. Gow (1994) showed that for any symmetric
group Sn , where is a prime, the number of Sylow -subgroups is ( p - 2)!. Hence
we target on finding the number of the Sylow p -subgroup for any symmetric group,
which will be given by;
( 1)!
( 2)!
0
m
p
n
Where is the number of Sylow p-subgroups of any symmetric group.