Applications of the Class Equation: Sylow's First Theorem and Probability in Group Theory

Applications of the Class Equation: Sylow’s First Theorem and Probability in Group Theory by Alexandre Gueganic ’19, Mathematics Colloquium, Wednesday, February 6, 1-1:45 pm, Stetson Court Classroom 101

Abstract:  The conjugacy class of a group forms an equivalence relation which permits the partitioning of a finite group G in the conjugacy classes of every elements of the finite group. From this observation, we provide a theorem linking the order of a conjugacy class of an element of G and the order of the centralizer of the same element of G. A direct corollary of this theorem yields the class equation. One of the applications of the class equation is to calculate the probability for two randomly chosen elements in a non-Abelian group to commute. Another strong theorem that we will discover deriving from the class equation is Sylow’s first theorem which proves that the converse of the Lagrange Theorem holds for every prime-power finite groups.