Metacyclic Group

There are two definitions of a metacyclic group.

1. A metacyclic group is a group G such that both its commutator subgroup G^' and the quotient group G/G^' are cyclic (Rose 1994, p. 247).

2. A group G is metacyclic if it has a cyclic normal subgroup L such that the quotient group G/L is also cyclic (Rose 1994, p. 56).

In general, a group may be metacyclic according to the second definition and fail the first one. For example, the quaternion group Q_8 has a normal cyclic subgroup of order 4, thus it satisfies definition (2). On the other hand, the commutant Q_8^' consists of two elements {1,-1}, the quotient Q_8/Q_8^' is isomorphic to the finite group C2×C2, and thus the group is not cyclic.

The first definition is more classical, but nowadays essentially all algebraists use the second definition, which is the one used in the remainder of this article.

Metacyclic groups are solvable and have a composition series of length two.

A complete classification of finite metacyclic groups has been given by Hempel (2000). The metacyclic groups are all generated by two elements which are subject to three relations depending on several numerical parameters. As a special, consider the groups generated by two elements alpha and beta such that

 alpha^n=1,    beta^m=1,    beta^(-1)alphabeta=alpha^p,

where p^m=1 (mod n). For n=p+1 and m=2 this is the definition of the dihedral group D_n, where alpha is the rotation of 2pi/n around the center and beta is the reflection around the axis of symmetry of a regular n-gon.

Dihedral groups are metacyclic, as is every subgroup and every quotient group of a metacyclic group (Rose 1994, p. 56).

A group is always metacyclic if its Sylow p-subgroups are all cyclic (Scott 1987, p. 356; Rose 1994, pp. 246-247). In particular, it follows that every group of squarefree order is metacyclic.

See also

Cyclic Group

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by Ashot Minasyan

Explore with Wolfram|Alpha


Alonso, J. "Groups of Square-Free Order, an Algorithm." Math. Comput. 30, 632-637, 1976.Hall, M. The Theory of Groups. Providence, RI: Chelsea, p. 146, 1976.Hempel, C. E. "Metacyclic Groups." Commun. Algebra 28, 3865-3897, 2000.Mac Lane, S. and Birkhoff, G. Algebra, rd ed. New York: Macmillan, p. 462, 1967.Rose, J. S. A Course on Group Theory. New York: Dover, 1994.Scott, W. R. Group Theory. New York: Dover, 1987.

Referenced on Wolfram|Alpha

Metacyclic Group

Cite this as:

Barile, Margherita; Minasyan, Ashot; and Weisstein, Eric W. "Metacyclic Group." From MathWorld--A Wolfram Web Resource.

Subject classifications