charlamov said:How would you prove that [itex] < x_1,x_2 ... | [ x_i , x_j ] =1, i,j \in N , x_1 ^ p = 1, x_{i+1} ^p = x_i , i \in N > [/itex] is presentation of [itex] Z_{p^ \infinity} [/itex]
A Prüfer group, also known as a cyclic group, is an infinite abelian group that is generated by a single element.
A Prüfer group is often represented using the notation Cn, where n is the order of the group. It can also be represented using the cyclic group notation ⟨a⟩, where a is the generator of the group.
A Prüfer group has a simple structure, with all elements being powers of the generator. It is also a torsion group, meaning that every element has finite order.
The Prüfer group is isomorphic to the additive group of integers, meaning that it shares the same algebraic structure as the integers. However, it is an infinite group, while the integers are finite.
The Prüfer group has applications in various areas of mathematics, including number theory, algebraic geometry, and cryptography. It is also used in physics, particularly in the study of symmetries and crystallography.