Sylow theory

Tema ovog završnog rada je objasniti sto su to Sylowljevi teoremi, što su Sylowljeve podgrupe i koliko ih ima te koliki je njihov red. Započet ćemo definicijom Sylowljeve podgrupe, a nakon iskazivanja Sylowljevih teorema reći ćemo nesto o posljedicama istih. Zatim ćemo objasniti tranzitivno djelovanje podgrupe te tako doći do Frattinijevog teorema. Bitno pitanje u pogledu konačnih grupa je pitanje prostosti koje ćemo pojasniti u jednome od poglavlja kao i broj elemenata prostog reda. Rad ćemo završiti Schur-Zassenhausovim teoremom koji nam govori o komplementima normalnih podgrupa.The topic of this nal paper is to explain what exactly are Sylow Theorems,what are Sylow subgroups and how many of them are there as well as which is their order. We will start with a denition of a Sylow subgroup and after state of Sylow Theorems we will say something about the consequences of such. Then, we will explain transitive action of a subgroup as well as how to reach to the Frattini Theorem. An important question with respect to nal subgroups is the question of simplycity as well as the number of elements of a prime order which will be explained in one of the upcoming chapters. This paper will be concluded with Schur-Zassenhaus Theorem which gives us an insight in complements of normal subgroups