Z-group

:Usage: {{harv|Suzuki|1955}}, {{harv|Bender|Glauberman|1994|p=2}}, {{MR|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|1959|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|1959|loc=Th. 9.4.3}}:

:$G(m,n,r)\; =\; \backslash langle\; a,b\; |\; a^m\; =\; b^n\; =\; 1,\; bab^\{-1\}\; =\; a^r\; \backslash rangle$, where mn is the order of G(m,n,r), the greatest common divisor, gcd((r-1)n, m) = 1, and rⁿ ≡ 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.

: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}}.

{{main|Zassenhaus group}}

: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(2^2k+1), for k any positive integer {{harv|Suzuki|1961}}.

• {{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 | mr=1311244 | year=1994 | volume=188}}
• {{Citation | last1=Çelik | first1=Özdem | title=On the character table of Z-groups | mr=0470050 | year=1976 | journal=Mitteilungen aus dem Mathematischen Seminar Giessen | issn=0373-8221 | pages=75–77}}
• {{citation | last=Hall | first=Marshall Jr. | title=The Theory of Groups | year=1959 | author-link=Marshall Hall (mathematician) | location=New York | publisher=Macmillan |mr=103215}}
• {{Citation | last1=Hall | first1=Philip | author1-link=Philip Hall | title=The construction of soluble groups | mr=0002877 | year=1940 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=182 | pages=206–214| doi=10.1515/crll.1940.182.206 | s2cid=118354698 }}
• {{Citation | last1=Kurosh | first1=A. G. | title=The theory of groups | publisher=Chelsea | location=New York | mr=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 (mathematician) | title=On finite groups with cyclic Sylow subgroups for all odd primes | mr=0074411 | year=1955 | journal=American Journal of Mathematics | issn=0002-9327 | volume=77 | pages=657–691 | doi=10.2307/2372591 | jstor=2372591 | issue=4}}
• {{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=Finite groups with nilpotent centralizers | mr=0131459 | year=1961 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=99 | pages=425–470 | doi=10.2307/1993556 | jstor=1993556 | issue=3| doi-access=free }}
• {{Citation | last1=Wonenburger | first1=María J.| author1-link=Maria Wonenburger | title=A generalization of Z-groups | mr=0393229 | year=1976 | journal=Journal of Algebra | issn=0021-8693 | volume=38 | issue=2 | pages=274–279 | doi=10.1016/0021-8693(76)90219-2| doi-access= }}
• {{Citation | last1=Zassenhaus | first1=Hans | author1-link=Hans Zassenhaus | title=Über endliche Fastkörper | language=German | year=1935 | journal=Abh. Math. Sem. Univ. Hamburg | volume=11 | pages=187–220 | doi=10.1007/BF02940723| s2cid=123632723 }}

Category:Infinite group theory

Category:Finite groups

Category:Properties of groups