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单词 MetacyclicGroup
释义

metacyclic group


Definition

A metacyclic groupMathworldPlanetmath is a group G that possesses a normal subgroupMathworldPlanetmath N such that N and G/N are both cyclic.

Examples

  • All cyclic groupsMathworldPlanetmath, and direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of two cyclic groups.

  • All dihedral groupsMathworldPlanetmath (including the infinite dihedral group).

  • All finite groupsMathworldPlanetmath whose Sylow subgroups are cyclic (and so, in particular, all finite groups of squarefree (http://planetmath.org/SquareFreeNumber) order).

Properties

SubgroupsMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) and quotientsPlanetmathPlanetmath (http://planetmath.org/QuotientGroup) of metacyclic groups are also metacyclic.

Metacyclic groups are obviously supersolvable, with Hirsch length at most 2.

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更新时间:2025/5/5 0:02:52