High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! 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: 1.In the study of finite groups, a Z-group is a finite groups whose Sylow subgroups are all cyclic. 2. in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. 3. occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. 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 Zyklische 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.

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