Z-group

Last updated August 19, 2023

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 $\mathbb {Z}$ -group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementarily equivalent to the integers $(\mathbb {Z} ,+,<)$ . 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

Usage: ( Suzuki 1955 ), ( Bender & Glauberman 1994 , p. 2), MR 0409648, ( Wonenburger 1976 ), ( 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 ( 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 , 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 ( Hall 1959 , Th. 9.4.3):

G(m,n,r)=\langle a,b|a^{m}=b^{n}=1,bab^{-1}=a^{r}\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 ( 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 ).

Related Research Articles

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In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C_n, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.

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In mathematics, specifically group theory, a subgroup series of a group $is a chain of subgroups:$

In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson.

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In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence

In mathematics, the Brauer–Suzuki theorem, proved by Brauer & Suzuki (1959), Suzuki (1962), Brauer (1964), states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a center of order 2. In particular, such a group cannot be simple.

In mathematics, George Glauberman's Z* theorem is stated as follows:

Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).

In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings, it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal.

In mathematical group theory, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power of p. In other words the group is a semidirect product of the normal p-complement and any Sylow p-subgroup. A group is called p-nilpotent if it has a normal p-complement.

In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian p-groups, the so-called, regular p-groups. The relationship between power and commutator structure forms a central theme in the modern study of p-groups, as exemplified in the work on uniformly powerful p-groups.

In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to. The focal subgroup theorem relates the ideas of transfer and fusion such as described in. Various applications of these ideas include local criteria for p-nilpotence and various non-simplicity criteria focussing on showing that a finite group has a normal subgroup of index p.

In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.

In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.

In mathematics, the Walter theorem, proved by John H. Walter, describes the finite groups whose Sylow 2-subgroup is abelian. Bender (1970) used Bender's method to give a simpler proof.

References

Bender, Helmut; Glauberman, George (1994), Local analysis for the odd order theorem, London Mathematical Society Lecture Note Series, vol. 188, Cambridge University Press, ISBN 978-0-521-45716-3, MR 1311244
Çelik, Özdem (1976), "On the character table of Z-groups", Mitteilungen aus dem Mathematischen Seminar Giessen: 75–77, ISSN 0373-8221, MR 0470050
Hall, Marshall Jr. (1959), The Theory of Groups, New York: Macmillan, MR 0103215
Hall, Philip (1940), "The construction of soluble groups", Journal für die reine und angewandte Mathematik, 182: 206–214, ISSN 0075-4102, MR 0002877
Kurosh, A. G. (1960), The theory of groups, New York: Chelsea, MR 0109842
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, JSTOR 2372591, MR 0074411
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, JSTOR 1993556, MR 0131459
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, MR 0393229
Zassenhaus, Hans (1935), "Über endliche Fastkörper", Abh. Math. Sem. Univ. Hamburg (in German), 11: 187–220, doi:10.1007/BF02940723, S2CID 123632723

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