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Reist, Benjamin Martin

Mathematics

2005

MS

Let n be a natural number and let G be a finite group possessing an element of order n. An order set in G is the set of all elements of order n in G, denoted by On (G), or simply On when G is clear from the context. Thus an order set is the union of the conjugacy classes of G whose elements have order n. We say an order set is a perfect order set if its cardinality divides the order of G. An order set consisting of a single conjugacy class is necessarily perfect, but one comprised of multiple classes may or may not be perfect even if all conjugacy classes of elements of order n have the same size. A perfect order group is a finite group in which all of its order sets are perfect. The Perfect Order Set Conjecture (POS-Conjecture) states that no finite simple group other than the cyclic group Z₂ is a perfect order group. Note that there are nonsimple perfect order groups; examples include: S₃, SL₂(3) and SL₂ (5) (in the first group each On is a single conjugacy class-hence its size divides the group order-but in the latter two groups there are perfect order sets that are unions of two classes). The POS-conjecture has been verified for the cyclic groups of prime order, the simple alternating groups, and the twenty-six sporadic simple groups. This thesis proves the POS-conjecture for the family of finite simple groups of exceptional Lie type F₄(q). In conclusion we conjecture that this proof can be generalized to prove the POS-conjecture for all of the families of finite simple groups of exceptional Lie type.