Formation (group theory)

In group theory, a branch of mathematics, 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. Gaschütz (1962) introduced formations to unify the theory of Hall subgroups and Carter subgroups of finite solvable groups.

Some examples of formations are the formation of p-groups for a prime 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

${\displaystyle 1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1\ }$

A and C are in M if and only if B is in M.^[1]

A full formation is a Melnikov formation which is also closed under taking subgroups.^[1]

An almost full formation is one which is closed under quotients, direct products and subgroups, but not necessarily extensions. The families of finite abelian groups and finite nilpotent groups are almost full, but neither full nor Melnikov.^[2]

Schunck classes

A Schunck class, introduced by

References

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  MR 2241927
• Doerk, Klaus; Hawkes, Trevor (1992), Finite soluble groups, de Gruyter Expositions in Mathematics, vol. 4, Berlin: Walter de Gruyter & Co.,
  MR 1169099
• Fried, Michael D.; Jarden, Moshe (2004), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11 (2nd revised and enlarged ed.),
  Zbl 1055.12003
• Gaschütz, Wolfgang (1962), "Zur Theorie der endlichen auflösbaren Gruppen",
  MR 0179257
• OCLC 527050
• Schunck, Hermann (1967), "H-Untergruppen in endlichen auflösbaren Gruppen",
  MR 0209356