| Identity Component |
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| CATEGORIES ABOUT IDENTITY COMPONENT | |
| topological groups | |
| lie groups | |
| mathematical components | |
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The identity component ''G''0 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 It follows that ''G''0 is normal in ''G''. It is not always true that ''G''0 is Open 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 . |
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