In mathematics, the identity component of a topological group ''G'' is the connected component ''G''0 that contains the identity element ''e''. The identity component ''G''0 is a closed set, normal subgroup of ''G''. It is closed since components are always closed. It is a subgroup since multiplication and inversion are continuous (topology). Moreover, for any continuous automorphism ''a'' of ''G'' we have :''a''(''G''0) = ''G''0. It follows that ''G''0 is normal in ''G''. It is not always true that ''G''0 is open set in ''G''. In fact, we may have ''G''0 = {''e''}, in which case ''G'' is totally disconnected. However, if ''G'' is a Lie group then ''G''0 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 ''G''0 is clopen. The quotient group ''G''/''G''0 is called the group of components 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. Topological groups Lie groups
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Identity component
In mathematics, the identity component of a topological group ''G'' is the connected component ''G''0 that contains the identity element ''e''. The identity component ''G''0 is a closed set, normal subgroup of ''G''. It is closed since components are always closed. It is a subgroup since multiplication and inversion are continuous (topology). Moreover, for any continuous automorphism ''a'' of ''G'' we have :''a''(''G''0) = ''G''0. It follows that ''G''0 is normal in ''G''. It is not always true that ''G''0 is open set in ''G''. In fact, we may have ''G''0 = {''e''}, in which case ''G'' is totally disconnected. However, if ''G'' is a Lie group then ''G''0 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 ''G''0 is clopen. The quotient group ''G''/''G''0 is called the group of components 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. Topological groups Lie groups
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