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Given ''G'' and a normal subgroup ''N'', then ''G'' is a [[group extension]] of {{nowrap|''G''/''N''}} by ''N''. One could ask whether this extension is trivial or split; in other words, one could ask whether ''G'' is a [[direct product of groups|direct product]] or [[semidirect product]] of ''N'' and {{nowrap|''G''/''N''}}. This is a special case of the [[extension problem]]. An example where the extension is not split is as follows: Let ''G'' = '''Z'''<sub>4</sub> = {0, 1, 2, 3}, and ''N'' = {0, 2}, which is isomorphic to '''Z'''<sub>2</sub>. Then {{nowrap|''G''/''N''}} is also isomorphic to '''Z'''<sub>2</sub>. But '''Z'''<sub>2</sub> has only the trivial [[automorphism]], so the only semi-direct product of ''N'' and {{nowrap|''G''/''N''}} is the direct product. Since '''Z'''<sub>4</sub> is different from {{nowrap|'''Z'''<sub>2</sub> × '''Z'''<sub>2</sub>}}, we conclude that ''G'' is not a semi-direct product of ''N'' and {{nowrap|''G''/''N''}}.
== 리 군의 몫 ==
==Quotients of Lie groups==
If ''<math>G</math>'' is a가 [[Lie리 group군]] and이고 ''<math>N</math>''이 is정규적이고 a normal and closed폐쇄적인 경우(in단어의 the대수적 topological의미보다는 rather위상적인 than the algebraic sense of the word경우) [[Lie subgroup]] of ''<math>G</math>'',의 the리 quotient부분군이면, {{nowrap개행 금지||''<math>G</math>'' / ''<math>N</math>''}}도 is리 also a Lie group군이다. 이 In경우 this case, the original group원래 ''<math>G</math>'' has군은 the기본 structure공간 of a [[fiber bundle]] (specifically, a [[principal{{개행 bundle금지|principal ''<math>NG</math>''-bundle]]), with/ base space {{nowrap|''<math>GN</math>''}}와 /올 ''<math>N</math>''}}을 and가진 fiber[[올다발]](특히 [[주다발|주''<math>N</math>''.다발]]) The구조를 dimension of갖는다. {{nowrap개행 금지|''<math>G</math>'' / ''<math>N</math>''}}의 equals치수는 <math> \mathrm{dim}\ G - \mathrm{dim}\ N</math>과 같다.<ref>John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17</ref>
''<math>N</math>''이 닫히는 조건은 필수이다. 실제로 ''<math>N</math>''이 닫히지 않는다면 몫 공간은 [[T1 공간]]이 아닌데 이는 열린 집합에 의해 항등원으로부터 분리될 수 없는 몫에는 잉여류가 있기 때문이다. 따라서 [[하우스도르프 공간]]이 아니다.
Note that the condition that ''<math>N</math>'' is closed is necessary. Indeed, if ''<math>N</math>'' is not closed then the quotient space is not a [[T1-space]] (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a [[Hausdorff space]].
For비정규 a리 non-normal Lie subgroup부분군 ''<math>N</math>''의 경우, the왼쪽 잉여류 space공간 {{nowrap개행 금지|''<math>G</math>'' / ''<math>N</math>''}}은 of군이 left아니라 cosetsG가 is not a group, but simply a작용하는 [[differentiable매끄러운 manifold다양체]] on which ''<math>G</math>'' acts이다. 그러한 The result is known as a결과를 [[homogeneous동차 space공간]]이라고 한다.
==See also==
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