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.

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All cyclic groups, and direct products of two cyclic groups.
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All dihedral groups (including the infinite dihedral group).
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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 $2$ .
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