Group decomposition

Core idea

A group decomposition describes a group in terms of simpler groups and the maps that assemble them. The precise form depends on how the constituent subgroups interact.

Common forms

• An internal direct product expresses a group as a product of commuting normal subgroups with trivial intersection.
• An internal semidirect product relaxes the commutativity requirement and records an action of one factor on the other.
• A group extension describes a group $G$ through an exact sequence
  $$1\longrightarrow N\longrightarrow G\longrightarrow Q\longrightarrow 1.$$
  When the extension splits, $G$ is a semidirect product of $N$ by $Q$.

Structural results

The available decomposition and its uniqueness depend on the class of groups under consideration. For example, the classification of finite abelian groups gives a direct-product decomposition into cyclic prime-power groups, while the Krull–Remak–Schmidt theorem gives uniqueness results for certain decompositions into directly indecomposable factors.