nLab structure group

Idea

For $G$ a group and $P$ a $G$-torsor, $G$ is also called the structure group of $P$.

This usage is particularly common in the case that $G$ is a geometric group (e.g. a topological group or Lie group) and $P \to X$ a $G$-principal bundle.

More generally, and less tautologically, if $V \to X$ is a $G$-associated bundle then one says that $V$ has structure group $G$.

In particular when $V = T X$ is a tangent bundle with structure group $G$ one also says that $X$ is equipped with G-structure.

In physics, with these bundles understood as field bundles (cf. fiber bundles in physics) the structure group is called the gauge group.

Related concepts

• reduction of structure group, G-structure

• gauge group

• equivariance group