Sums of element orders in finite group

This thesis consist of five chapters. In first chapter, the history of element orders in finite groups and the studies in the literature are given. In second chapter, the basic definitions and theorems necessary in the following chapters are mentioned. In third chapter, the results obtained on the sum of element orders in finite groups are investigated. In fourth chapter, the results obtained on the sum of the element orders in the finite abelian groups are discussed. In last chapter, some results obtained from a formula which calculates an exact upper bound for sums of element orders non-cyclic finite groups are examined.