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# Identity component

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

Properties

The identity component *G*_{0} of a topological group *G* is a closed normal subgroup of *G*. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological group are continuous maps by definition. Moreover, for any continuous automorphism *a* of *G* we have

*a*(*G*_{0}) =*G*_{0}.

Thus, *G*_{0} is a characteristic subgroup of *G*, so it is normal.

The identity component *G*_{0} of a topological group *G* need not be open in *G*. In fact, we may have *G*_{0} = {*e*}, in which case *G* is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {*e*}; and therefore is a clopen set.

The identity path component may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if *G* is locally path-connected.

Component group

The quotient group *G*/*G*_{0} is called the **group of components** or **component group** of *G*. Its elements are just the connected components of *G*. The component group *G*/*G*_{0} is a discrete group if and only if *G*_{0} is open. If *G* is an affine algebraic group then *G*/*G*_{0} is actually a finite group.

One may similarly define the path component group as the group of path components (quotient of *G* by the identity path component), and in general the component group is a quotient of the path component group, but if *G* is locally path connected these groups agree. The path component group can also be characterized as the zeroth homotopy group, \( \pi_0(G,e). \)

Examples

- The group of non-zero real numbers with multiplication (
**R***,•) has two components and the group of components is ({1,−1},•). - Consider the group of units
*U*in the ring of split-complex numbers. In the ordinary topology of the plane {*z*=*x*+ j*y*:*x*,*y*∈**R**},*U*is divided into four components by the lines*y*=*x*and*y*= −*x*where*z*has no inverse. Then*U*_{0}= {*z*: |*y*| <*x*} . In this case the group of components of*U*is isomorphic to the Klein four-group.

References

Lev Semenovich Pontryagin, Topological Groups, 1966.

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