metacyclic group

Definition

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

•
All cyclic groups, and direct products of two cyclic groups.
•
All dihedral groups (including the infinite dihedral group).
•
All finite groups whose Sylow subgroups are cyclic (and so, in particular, all finite groups of squarefree (http://planetmath.org/SquareFreeNumber) order).

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

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

单词	MetacyclicGroup
释义	metacyclic group Definition A metacyclic group is a group $G$ that possesses a normal subgroup $N$ such that $N$ and $G/N$ are both cyclic. Examples • All cyclic groups, and direct products of two cyclic groups. • All dihedral groups (including the infinite dihedral group). • All finite groups whose Sylow subgroups are cyclic (and so, in particular, all finite groups of squarefree (http://planetmath.org/SquareFreeNumber) order). Properties Subgroups (http://planetmath.org/Subgroup) and quotients (http://planetmath.org/QuotientGroup) of metacyclic groups are also metacyclic. Metacyclic groups are obviously supersolvable, with Hirsch length at most $2$ .
随便看	uniform continuity UniformContinuityOfLipschitzFunctions uniform continuity of Lipschitz functions UniformContinuityOverLocallyCompactQuantumGroupoids uniform continuity over locally compact quantum groupoids UniformcontinuousRandomVariable uniform (continuous) random variable UniformConvergence uniform convergence UniformConvergenceOfIntegral uniform convergence of integral UniformConvergenceOnUnionInterval CanonicalModel canonical model CanonicalOrderingOnPairsOfOrdinals canonical ordering on pairs of ordinals CanonicalProjection canonical projection CanonicalQuantization canonical quantization CantorBendixsonDerivative Cantor-Bendixson derivative CantorBendixsonTheorem Cantor-Bendixson theorem CantorFunction