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 a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a formation is a
class of groups 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 subgroups and
Carter subgroups of
finite solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminat ...
s.
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 ...
s and
group extensions. Thus a Melnikov formation ''M'' has the property that for every
short exact sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
:
''A'' and ''C'' are in ''M''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''B'' is in ''M''.
A full formation is a Melnikov formation which is also closed under taking
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
s.
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 commu ...
s and finite
nilpotent groups are almost full, but neither full nor Melnikov.
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.
Notes
References
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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 , issn=0025-5874 , volume=97 , pages=326–330
Group theory