In mathematics, the identity component of a topological group G is the connected component G₀ that contains the identity element e.

The identity component G₀ is a closed, normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion are continuous maps. Moreover, for any continuous automorphism a of G we have

a(G₀) = G₀.

It follows that G₀ is normal in G.

It is not always true that G₀ is open in G. In fact, we may have G₀ = {e}, in which case G is totally disconnected. However, if G is a Lie group then G₀ is open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set. More generally, for any locally connected topological group the identity component G₀ is clopen.

The quotient group G/G₀ is called the group of components of G. Its elements are just the connected components of G. The component group G/G₀ is a discrete group if and only if G₀ is open. If G is an affine algebraic group then G/G₀ is actually a finite group.