About: 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. * 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 ordered groups, a Z-group or -group is a discretely ordered abelian group whose quotient over its minimal is divisible. Such groups are elementarily equivalent to the integers . 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.