Prüfer p-Group

n

1

2

3

4

5

prime p

2

3

5

7

11

circles/disks

The Prüfer

p

-group

(

∞

p

)

for a prime number

p

consists of all roots of unity of order

n

p

,

n∈

+



; that is,

(

∞

p

)=

2πim

n

p

e

:n∈

+



,m=1,…,

n

p



. The group, named after the mathematician Heinz Prüfer, is also known as the quasicyclic

p

-group.

(

∞

p

)

is an example of a countable

p

-group that is not the direct sum of groups of rank 1. The radii of the circles or disks for each element on the group in the plot decrease with

n

.