# 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**[math]\displaystyle{ \mathbb Z }[/math]-group**is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementarily equivalent to the integers [math]\displaystyle{ (\mathbb Z,+,\lt ) }[/math]. 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

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 (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 (Hall 1959) 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 (Hall 1959):

- [math]\displaystyle{ G(m,n,r) = \langle a,b | a^m = b^n = 1, bab^{-1} = 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 (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 (Hall 1940). Another generalization due to (Suzuki 1955) allows the Sylow 2-subgroup more flexibility, including dihedral and generalized quaternion groups.

## Group with a generalized central series

*Usage: (Robinson 1996), (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 (Robinson 1996).

## Special 2-transitive groups

*Usage: (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 (Suzuki 1961).

## References

- Bender, Helmut; Glauberman, George (1994),
*Local analysis for the odd order theorem*, London Mathematical Society Lecture Note Series,**188**,*Cambridge University Press*, ISBN 978-0-521-45716-3 - Çelik, Özdem (1976), "On the character table of Z-groups",
*Mitteilungen aus dem Mathematischen Seminar Giessen*: 75–77, ISSN 0373-8221 - Hall, Marshall Jr. (1959),
*The Theory of Groups*, New York: Macmillan - Hall, Philip (1940), "The construction of soluble groups",
*Journal für die reine und angewandte Mathematik***182**: 206–214, doi:10.1515/crll.1940.182.206, ISSN 0075-4102 - Kurosh, A. G. (1960),
*The theory of groups*, New York: Chelsea - Robinson, Derek John Scott (1996),
*A course in the theory of groups*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6 - Suzuki, Michio (1955), "On finite groups with cyclic Sylow subgroups for all odd primes",
*American Journal of Mathematics***77**(4): 657–691, doi:10.2307/2372591, ISSN 0002-9327 - Suzuki, Michio (1961), "Finite groups with nilpotent centralizers",
*Transactions of the American Mathematical Society***99**(3): 425–470, doi:10.2307/1993556, ISSN 0002-9947 - Wonenburger, María J. (1976), "A generalization of Z-groups",
*Journal of Algebra***38**(2): 274–279, doi:10.1016/0021-8693(76)90219-2, ISSN 0021-8693 - Zassenhaus, Hans (1935), "Über endliche Fastkörper" (in German),
*Abh. Math. Sem. Univ. Hamburg***11**: 187–220, doi:10.1007/BF02940723

Original source: https://en.wikipedia.org/wiki/Z-group.
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