  This is an overview of the idea of a crossed module. For a group, the triple
that consists of the group, its group of automorphisms, and the canonical
homomorphism from the group to its group of automorphisms constitutes a crossed
module. Crossed modules arise from the identities among the relations of the
presentation of a group, from the extension problem for groups and, more
generally, in low dimensional topology. Also, the (successful) attempt to
extend the idea of a normal extension of commutative fields to the realm of
non-commutative algebras leads to crossed modules. Crossed modules appear
implicitly in a forgotten paper by A. Turing which in principle settles the
extension problem for groups. Crossed modules make perfect sense for Lie
algebras.
