Group theory MOC

Group extension

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

1→𝐵↪𝐺↠𝐴→1

where 1 denotes the trivial group. Hence 𝐺 “covers” 𝐴 with the kernel 𝐵 ↪𝐺. Note that 𝐵 is necessarily a normal subgroup, giving the quotient 𝐺/𝐵 ≅𝐴 by the First isomorphism theorem. Two extensions 𝐺1,𝐺2 of 𝐴 by 𝐵 are said to be equivalent iff there exists an isomorphism such that the following diagram commutes

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

Classification

Consider an extension 1 →𝐵 →𝐺 𝑝↠𝐴 →1.

1. Iff 𝐵 is abelian, one speaks of an abelian extension
2. Iff 𝐵 ↪𝐺 is central, one speaks of a central extension.
3. Iff 𝐺 ≅𝐵 ⋊𝐴 (Semidirect product), one speaks of a split extension, equivalently 𝑝 is split epic.
4. Iff 𝔤 ≅𝔟 ×𝔞 (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)

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Footnotes

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