A metacyclic group consists of two cyclic layers and is one of the simplest nonabelian extension classes. Such groups admit two-generator presentations with a small number of parameters. [@isaacs2008]
Definition
Metacyclic does not mean a direct product of two cyclic groups and does not require the extension to split.
Core result
Quotients preserve cyclic images. For a subgroup \(H\), use cyclic normal \(H\cap N\) and embed \(H/(H\cap N)\) in cyclic \(G/N\). The transfer consequence excludes the nonmetacyclic quotient \(C_p\wr C_p\).
Structural properties
- A finite metacyclic group needs at most two generators.
- Cyclic and dihedral groups are standard examples.
- Metacyclic \(p\)-groups can have arbitrarily large nilpotency class.
- Splitting depends on presentation parameters and coprimeness.
- Closure under subgroups and quotients makes forbidden-quotient theorems effective.
Metacyclicity is an extension property, broader than abelianness but narrower than general two-generated solvable groups.
Example and boundary
Knowledge network
Split extensions describe common constructions, Yoshida uses the forbidden wreath quotient, and regular p-groups provide another sufficient class.
Proof and sources
Closure, the nonmetacyclic wreath quotient, and Huppert's transfer result are Theorems 10.12-10.15. [@isaacs2008]
