Z-group

In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of groups:

In the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic. The Z originates both from the German and from their classification in (Zassenhaus 1935). In many standard textbooks these groups have no special name, other than metacyclic groups, but that term is often used more generally today. See metacyclic group for more on the general, modern definition which includes non-cyclic p-groups; see (Hall 1969, Th. 9.4.3) for the stricter, classical definition more closely related to Z-groups.

Every group whose Sylow subgroups are cyclic is itself metacyclic, so supersolvable. In fact, such a group has a cyclic derived subgroup with cyclic maximal abelian quotient. Such a group has the presentation (Hall 1969, Th. 9.4.3):

The character theory of Z-groups is well understood (Çelik 1976), as they are monomial groups.

The derived length of a Z-group is at most 2, so Z-groups may be insufficient for some uses. A generalization due to Hall are the A-groups, those groups with abelian Sylow subgroups. These groups behave similarly to Z-groups, but can have arbitrarily large derived length (Hall 1940). Another generalization due to (Suzuki 1955) allows the Sylow 2-subgroup more flexibility, including dihedral and generalized quaternion groups.

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