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# Identity component

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.

03-10-2013 05:06:04