Metacyclic group

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

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

gollark: Does anyone know what the weather implications of locally blotting out the sun with a giant space mirror would be?
gollark: Soon: the thunderstorms and fire accidentally open a portal to hell.
gollark: I don't think the terms mean the weird cold-war-y definitions when most people actually use them.
gollark: https://en.wikipedia.org/wiki/Second_World
gollark: What, *gone*? Very <:MildPanic:579802652888662018>.

References

  • A. L. Shmel'kin (2001) [1994], "Metacyclic group", Encyclopedia of Mathematics, EMS Press


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