In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a formation is a
class of groups
A class of groups is a set theoretical collection of groups satisfying the property that if ''G'' is in the collection then every group isomorphic to ''G'' is also in the collection. This concept arose from the necessity to work with a bunch of gr ...
closed under taking images and such that if ''G''/''M'' and ''G''/''N'' are in the formation then so is ''G''/''M''∩''N''. introduced formations to unify the theory of
Hall subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist .
Definitions
A Hall divisor (also called a unitary divisor) of a ...
s and
Carter subgroup
In mathematics, especially in the field of group theory, a Carter subgroup of a finite group ''G'' is a self-normalizing subgroup of ''G'' that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post ...
s of
finite solvable groups.
Some examples of formations are the formation of
''p''-groups for a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'', the formation of π-groups for a set of primes π, and the formation of
nilpotent groups.
Special cases
A Melnikov formation is closed under taking
quotients,
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
s and
group extensions. Thus a Melnikov formation ''M'' has the property that for every
short exact sequence
:
''A'' and ''C'' are in ''M''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
''B'' is in ''M''.
[Fried & Jarden (2004) p.344]
A full formation is a Melnikov formation which is also closed under taking
subgroups.
[
An almost full formation is one which is closed under quotients, direct products and subgroups, but not necessarily extensions. The families of finite ]abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s and finite nilpotent groups are almost full, but neither full nor Melnikov.[Fried & Jarden (2004) p.542]
Schunck classes
A Schunck class, introduced by , is a generalization of a formation, consisting of a class of groups such that a group is in the class if and only if every primitive factor group is in the class. Here a group is called primitive if it has a self- centralizing normal abelian subgroup.
References
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*
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*
*
*{{Citation , last1=Schunck , first1=Hermann , title=H-Untergruppen in endlichen auflösbaren Gruppen , doi=10.1007/BF01112173 , mr=0209356 , year=1967 , journal=Mathematische Zeitschrift
''Mathematische Zeitschrift'' (German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag.
It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard ...
, issn=0025-5874 , volume=97 , pages=326–330
Group theory