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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 groups whose Sylow subgroups are all cyclic. * in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. * occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. ==Groups whose Sylow subgroups are cyclic== :''Usage: , , , , '' 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 . 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 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 : :, where ''mn'' is the order of ''G''(''m'',''n'',''r''), the greatest common divisor, gcd((''r''-1)''n'', ''m'') = 1, and ''r''''n'' ≡ 1 (mod ''m''). The character theory of Z-groups is well understood , 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 . Another generalization due to allows the Sylow 2-subgroup more flexibility, including dihedral and generalized quaternion groups. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Z-group」の詳細全文を読む スポンサード リンク
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