Group theory MOC

Group extension

Let be groups. An extension of by is a group together with a short exact sequence group

where denotes the trivial group. Hence “covers” with the kernel . Note that is necessarily a normal subgroup, giving the quotient by the First isomorphism theorem. Two extensions of by are said to be equivalent iff there exists an isomorphism such that the following diagram commutes

Following the ATLAS¹, we adopt the notation for an unspecified extension of by (so that is normal), and denote a non-split extension of by with .

Classification

Consider an extension .

1. Iff is abelian, one speaks of an abelian extension
2. Iff is central, one speaks of a central extension.
3. Iff (Semidirect product), one speaks of a split extension, equivalently is split epic.
4. Iff (Direct product of groups), one speaks of a trivial extension.

Proof of equivalence in 3.

proof

See also

• Lie algebra extension (the structure of that Zettel deliberately mirrors this one)

---

tidy | en | sembr

Footnotes

1. 1985. Atlas of finite groups: Maximal subgroups and ordinary characters for simple groups ↩