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 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: harv|Suzuki|1955, harv|Bender|Glauberman|1994|p=2, MathSciNet | id = 0409648, harv|Wonenburger|1976, harv|Çelik|1976"

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 harv|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 harv|Hall|1969|loc=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 harv|Hall|1969|loc=Th. 9.4.3::$G(m,n,r)\; =\; langle\; a,b\; |\; a^n\; =\; b^m\; =\; 1,\; a^b\; =\; a^r\; angle$, 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 harv|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 harv|Hall|1940. Another generalization due to harv|Suzuki|1955 allows the Sylow 2-subgroup more flexibility, including dihedral and generalized quaternion groups.

Group with a generalized central series

:"Usage: harv|Robinson|1996, harv|Kurosh|1960"

The definition of central series used for Z-group is somewhat technical. A series of "G" is a collection "S" of subgroups of "G", linearly ordered by inclusion, such that for every "g" in "G", the subgroups "A"_"g" = ∩ { "N" in "S" : "g" in "N" } and "B"_"g" = ∪ { "N" in "S" : "g" not in "N" } are both in "S". A (generalized) central series of "G" is a series such that every "N" in "S" is normal in "G" and such that for every "g" in "G", the quotient "A"_"g"/"B"_"g" is contained in the center of "G"/"B"_"g". A Z-group is a group with such a (generalized) central series. Examples include the hypercentral groups whose transfinite upper central series form such a central series, as well as the hypocentral groups whose transfinite lower central series form such a central series harv|Robinson|1996.

pecial 2-transitive groups

:"Usage: harv|Suzuki|1961"

A (Z)-group is a group faithfully represented as a doubly-transitive permutation group in which no non-identity element fixes more than two points. A (ZT)-group is a (Z)-group that is of odd degree and not a Frobenius group, that is a Zassenhaus group of odd degree, also known as one of the groups PSL(2,2^"k"+1) or Sz(2^2"k"+1), for "k" any positive integer harv|Suzuki|1961.

References

*Citation | last1=Bender | first1=Helmut | last2=Glauberman | first2=George | author2-link=George Glauberman | title=Local analysis for the odd order theorem | publisher=Cambridge University Press | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-45716-3 | id=MathSciNet | id = 1311244 | year=1994 | volume=188
*Citation | last1=Çelik | first1=Özdem | title=On the character table of Z-groups | id=MathSciNet | id = 0470050 | year=1976 | journal=Mitteilungen aus dem Mathematischen Seminar Giessen | issn=0373-8221 | pages=75–77
*Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The construction of soluble groups | id=MathSciNet | id = 0002877 | year=1940 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=182 | pages=206–214
*Citation | last1=Kurosh | first1=A. G. | title=The theory of groups | publisher=Chelsea | location=New York | id=MathSciNet | id = 0109842 | year=1960
*Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996
*Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=On finite groups with cyclic Sylow subgroups for all odd primes | id=MathSciNet | id = 0074411 | year=1955 | journal=American Journal of Mathematics | issn=0002-9327 | volume=77 | pages=657–691
*Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=Finite groups with nilpotent centralizers | id=MathSciNet | id = 0131459 | year=1961 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=99 | pages=425–470
*Citation | last1=Wonenburger | first1=María J. | title=A generalization of Z-groups | id=MathSciNet | id = 0393229 | year=1976 | journal=Journal of Algebra | issn=0021-8693 | volume=38 | issue=2 | pages=274–279
*Citation | last1=Zassenhaus | first1=Hans | author1-link=Hans Julius Zassenhaus | title=Über endliche Fastkörper | language=German | year=1935 | journal=Abh. Math. Semin. Hamb. Univ. | volume=11 | pages=187–220