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单词 QuasicyclicGroup
释义

quasicyclic group


Let p be a prime number.The p-quasicyclic group (or Prüfer p-group, or p group) is the p-primary componentPlanetmathPlanetmath of /,that is, the unique maximal p-subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/PGroup4) of /.Any group (http://planetmath.org/Group) isomorphicPlanetmathPlanetmathPlanetmath to this will also be called a p-quasicyclic group.

The p-quasicyclic group will be denoted by (p).Other notations in use include [p], /p,p and Cp.

(p) may also be defined in a number of other (equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) ways(again, up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath):

  • (p) isthe group of all pn-th complex roots of 1, for n.

  • (p) isthe injective hull of /p (viewing abelian groupsMathworldPlanetmath as -modules (http://planetmath.org/Module)).

  • (p) is the direct limitMathworldPlanetmath of the groups /pn.

A quasicyclic group (or Prüfer group) isa group that is p-quasicyclic for some prime p.

The subgroup (http://planetmath.org/Subgroup) structureMathworldPlanetmath of (p) is particularly simple:all proper subgroupsMathworldPlanetmath are finite and cyclic,and there is exactly one of order pn for each non-negative integer n.In particular,this means that the subgroups are linearly ordered by inclusion,and all subgroups are fully invariant.The quasicyclic groups arethe only infinite groups with a linearly ordered subgroup lattice.They are alsothe only infiniteMathworldPlanetmath solvable groupsMathworldPlanetmath whose proper subgroups are all finite.

Quasicyclic groups are locally cyclic, divisible (http://planetmath.org/DivisibleGroup) and co-Hopfian.

Every infinite locally cyclic p-group is isomorphic to (p).

Titlequasicyclic group
Canonical nameQuasicyclicGroup
Date of creation2013-03-22 15:35:22
Last modified on2013-03-22 15:35:22
Owneryark (2760)
Last modified byyark (2760)
Numerical id19
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F50
Classificationmsc 20K10
Synonymquasi-cyclic group
SynonymPrüfer group
Definesquasicyclic
Definesquasi-cyclic
DefinesPrüfer p-groupMathworldPlanetmath
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更新时间:2025/5/4 2:11:49