Superconducting radio frequency

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193.62.111.10: /* Physics of SRF cavities */ Capitalized Cooper pairs

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 group]]s, a '''Z-group''' is a finite groups whose [[Sylow subgroup]]s are all [[cyclic group|cyclic]].
* in the study of [[infinite group]]s, 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}}, {{MR|0409648}}, {{harv|Wonenburger|1976}}, {{harv|Çelik|1976}}''

In the study of [[finite group]]s, a '''Z-group''' is a finite group whose [[Sylow subgroup]]s are all [[cyclic group|cyclic]]. The Z originates both from the German [[:de:Zyclische gruppe|''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-group|''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 group|metacyclic]], so [[supersolvable group|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}}:
:<math>G(m,n,r) = \langle a,b | a^n = b^m = 1, a^b = a^r \rangle </math>, 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 group]]s.

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-group]]s, those groups with [[abelian group|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 group|dihedral]] and [[generalized quaternion group]]s.

==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 group]]s whose transfinite [[upper central series]] form such a central series, as well as the [[hypocentral group]]s whose transfinite lower central series form such a central series {{harv|Robinson|1996}}.

==Special 2-transitive groups==
{{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 [[projective special linear group|PSL(2,2''k''+1)]] or [[Group of Lie type#Suzuki–Ree groups|Sz(22''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 | 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, jr | first=Marshall | authorlink=Marshall Hall (mathematician) | title=The Theory of Groups | year=1969 | publisher=Macmillan | location=New York }}
*{{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}}
*{{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 | 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 | 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}}
*{{Citation | last1=Wonenburger | first1=María J. | 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}}
*{{Citation | last1=Zassenhaus | first1=Hans | author1-link=Hans Zassenhaus | title=Über endliche Fastkörper | language=German | year=1935 | journal=Abh. Math. Semin. Hamb. Univ. | volume=11 | pages=187–220 | doi=10.1007/BF02940723}}

[[Category:Infinite group theory]]
[[Category:Finite groups]]
[[Category:Properties of groups]]

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