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'''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]]

[[Category:Abstract algebra]]
[[Category:Group theory]]

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