an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some by Cayley's theorem). In addition it is both affine and projective. Thus, in particular for classification purposes, it is natural to restrict statements to connected algebraic group. Algebraic groups over local fields and Lie groups If the field is a local field (for instance the real or complex numbers, or a p-adic field) and is a -group then the group is endowed with the analytic topology coming from any embedding into a projective space as a quasi-projective variety. This is a group topology, and it makes into a topological group. Such groups are important examples in the general theory of topological groups. If or then this makes into a Lie group. Not all Lie groups can be obtained via this procedure, for example the universal cover of SL2(R), or the quotient of the Heisenberg group by a infinite normal discrete subgroup. An algebraic group over the real or complex numbers may have closed subgroups (in the analytic topology) which do not have the same connected component of the identity as any algebraic subgroup. Coxeter groups and algebraic groups There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is , and the number of elements of the general linear group over a finite field is the q-factorial ; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element. See also Character variety Borel subgroup Tame group Morley rank Cherlin–Zilber conjecture Adelic algebraic group Pseudo-reductive group References Milne, J. S., Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups Further reading Algebraic groups and their Lie algebras by Daniel Miller Properties of groups