metacyclic

Metacyclic is a term used in group theory to describe a class of groups that are built from cyclic pieces. A group G is called metacyclic if it contains a cyclic normal subgroup N such that the quotient G/N is cyclic. Equivalently, G has a cyclic normal subgroup with a cyclic quotient, making it a cyclic-by-cyclic group.

In the finite case, this means there is an exact sequence 1 → C_m → G → C_n → 1 for some integers m and n, where C_k denotes a cyclic group of order k. When the extension splits, G is isomorphic to a semidirect product C_m ⋊_φ C_n, determined by an action φ of C_n on C_m. Writing a as a generator of C_m and b as a preimage of the generator of C_n, one often encounters the presentation G = ⟨a, b | a^m = 1, b^n = 1, b a b^{-1} = a^r⟩ with r^n ≡ 1 (mod m). The integer r encodes the action of C_n on C_m. If the extension is not split, b^n may lie in ⟨a⟩, yielding a non-split metacyclic group.

Examples include cyclic groups (n = 1), dihedral groups D_{2m} (n = 2 and r = −1 modulo m), and generalized quaternion groups Q_{4m}, all of which have a normal cyclic subgroup with cyclic quotient. The infinite dihedral group D_∞ is a metacyclic example in the infinite setting.

Properties and scope: finite metacyclic groups are solvable and in fact metabelian (their commutator subgroup is abelian). Not every finite group is metacyclic; for instance, S_4 and many larger groups are not. Metacyclic groups form a natural bridge between cyclic groups and more complex non-abelian structures, capturing a wide range of two-generator, cyclic-structured formations.