A semipermutable subgroup in S4 is a subgroup of the symmetric group S4 in which every element commutes with some element of the subgroup. In other words, the subgroup and its elements have a specific relationship with the elements of the symmetric group S4.
A regular subgroup in S4 is a subgroup in which every element commutes with every other element. In contrast, a semipermutable subgroup in S4 only has the property that every element commutes with some element of the subgroup, but not necessarily with every other element.
Semipermutable subgroups in S4 have been studied extensively in group theory and have many applications in various branches of mathematics, including algebra and combinatorics. They also have connections to other mathematical structures, such as Latin squares and group factorizations.
One example of a semipermutable subgroup in S4 is the subgroup generated by the elements (12) and (34). This subgroup consists of the elements (12), (34), (13), (24), (14), (23), and the identity element (1). Each of these elements commutes with at least one other element in the subgroup.
Semipermutable subgroups in S4 have connections to other group properties, such as normal subgroups, abelian subgroups, and centralizers. They also have relationships with other concepts in group theory, such as nilpotent and solvable groups.