Finite Abelian Groups

Abstract

This chapter discusses the finite Abelian groups. Finite Abelian groups constitute an important chapter in algebra, which is practically covered by the Frobenius–Stickelberger main theorem. However, even today it still provides many interesting problems and this has led to its expansion recently by several new important studies. The chapter presents a theorem which states that every finite Abelian group is a direct product of cyclic groups of prime-power order (> 1). This theorem is also called the first main theorem for finite Abelian groups. If the order of a finite group G is divisible by a prime number p , then G contains at least one element of order p .