In this book we will consider two topics of the algebraic structures which concern the properties of finitely presented groups and semigroups. The first problem which we examine is the calculating of the Fibonacci length of two families of finitely presented groups. Also we get two applications of the Fibonaccilength on the classification of groups and graphs. These applications are theoretical results of this notion after all of its nice numerical results. The second problem which is investigated is a classification method for groups (the permutational property) which considered in 1987 by...