Trisha Shetty (Editor)

Metacyclic group

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In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence

Contents

1 → K → G → H → 1 ,

where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.

Properties

Metacyclic groups are both supersolvable and metabelian.

Examples

  • Any cyclic group is metacyclic.
  • The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups.
  • The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.)
  • Every finite group of squarefree order is metacyclic.
  • More generally every Z-group is metacyclic. A Z-group is a group whose Sylow subgroups are cyclic.

    • References

    Metacyclic group Wikipedia
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