Let be a prime number. The -quasicyclic group (or Prüfer -group, or group) is the -primary component of , that is, the unique maximal -subgroup (http://planetmath.org/PGroup4) of . Any group (http://planetmath.org/Group) isomorphic to this will also be called a -quasicyclic group.
The -quasicyclic group will be denoted by . Other notations in use include , , and .
A quasicyclic group (or Prüfer group) is a group that is -quasicyclic for some prime .
The subgroup (http://planetmath.org/Subgroup) structure of is particularly simple: all proper subgroups are finite and cyclic, and there is exactly one of order for each non-negative integer . In particular, this means that the subgroups are linearly ordered by inclusion, and all subgroups are fully invariant. The quasicyclic groups are the only infinite groups with a linearly ordered subgroup lattice. They are also the only infinite solvable groups whose proper subgroups are all finite.
Every infinite locally cyclic -group is isomorphic to .
|Date of creation||2013-03-22 15:35:22|
|Last modified on||2013-03-22 15:35:22|
|Last modified by||yark (2760)|