Group extension

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

1 \to 𝐵 ↪ 𝐺 ↠ 𝐴 \to 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 $𝐺 1, 𝐺 2$ 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 $𝐴 \cdot 𝐵$ .

Classification

Consider an extension $1 \to 𝐵 \to 𝐺 𝑝 ↠ 𝐴 \to 1$ .

Iff $𝐵$ is abelian, one speaks of an abelian extension
Iff $𝐵 ↪ 𝐺$ is central, one speaks of a central extension.
Iff $𝐺 ≅ 𝐵 ⋊ 𝐴$ (Semidirect product), one speaks of a split extension, equivalently $𝑝$ is split epic.
Iff $𝔤 ≅ 𝔟 \times 𝔞$ (Direct product of groups), one speaks of a trivial extension.

Proof of equivalence in 3.

#missing/proof

Group extension

Classification

See also

Footnotes