Identity component - Definition and Overview

In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e.

It follows immediately that C is a subgroup. It is usually written G^o. For any continuous automorphism a of G we have

a(G^o) = G^o.

Therefore, a fortiori, it is a normal subgroup. We may have G^o = {e}, in which case G is totally disconnected.

If G is a Lie group, G^o is open, since it contains a path-connected neighbourhood of {e}; and therefore is clopen. The quotient group G/G^o is then a discrete group. If G is an affine algebraic group, G/G^o is a finite group.