Metacyclic group

Metacyclic group is a concept in the field of abstract algebra, specifically within the study of group theory. A metacyclic group is a group that can be expressed as an extension of a cyclic group by another cyclic group. This means that a metacyclic group is built from two cyclic groups, one acting on the other in a specific manner. Understanding metacyclic groups is important for various areas of mathematics and its applications, including number theory, cryptography, and the study of symmetry in chemical and physical systems.

Definition

A group G is called metacyclic if there exist cyclic subgroups H and K of G such that H is normal in G, K is a subgroup of G, and the quotient group G/H is cyclic. In other words, G fits into an exact sequence of the form:

1 → H → G → G/H → 1

where H and G/H are both cyclic groups. The group G can then be described in terms of generators and relations involving these generators of H and G/H.

Examples

1. Every cyclic group is trivially metacyclic, as it can be considered as an extension of a cyclic group by the trivial group. 2. The dihedral groups, which are well-known in the context of symmetries of polygons, are metacyclic. A dihedral group of order 2n can be seen as an extension of a cyclic group of order n by a cyclic group of order 2. 3. Certain Galois groups that arise in the study of field extensions are metacyclic.

Properties

- Metacyclic groups are a generalization of cyclic groups. While all cyclic groups are metacyclic, the converse is not true. - The structure of metacyclic groups can be explicitly determined by the structure of its constituent cyclic groups and how one acts on the other. - Metacyclic groups play a role in the classification of groups of small order, contributing to the broader understanding of group theory.

Applications

Metacyclic groups find applications in various areas of mathematics and science. In number theory, they are involved in the study of Fermat's Last Theorem and cyclic Galois extensions. In cryptography, the structure of metacyclic groups is exploited in certain cryptographic protocols where the difficulty of solving group-related problems forms the basis of security. Additionally, in chemistry and physics, the symmetry properties of molecules and crystals can sometimes be described using metacyclic groups.

See Also

• Group theory
• Cyclic group
• Dihedral group
• Galois group