|
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 group (mathematics), groups: * in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic group, cyclic. * in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. * in the study of linearly ordered group, ordered groups, a Z-group or \mathbb Z-group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementary equivalence, elementarily equivalent to the integers (\mathbb Z,+,<). Z-groups are an alternative presentation of Presburger arithmetic. * 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 subgr ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersolvable Group
In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. Definition Let ''G'' be a group. ''G'' is supersolvable if there exists a normal series :\ = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_ \triangleleft H_s = G such that each quotient group H_/H_i \; is cyclic and each H_i is normal in G. By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a subnormal series with each quotient cyclic, but there is no requirement that each H_i be normal in G. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points, A_4, is solvable but not supersolvable. Basic Properties Some facts about supersolvable groups: * Supersol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |