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Orthogonal group
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== Of complex quadratic forms == Over the field {{math|'''C'''}} of [[complex number]]s, every non-degenerate [[quadratic form]] in {{mvar|n}} variables is equivalent to {{math|''x''{{sub|1}}{{sup|2}} + ... + ''x''{{sub|''n''}}{{sup|2}}}}. Thus, up to isomorphism, there is only one non-degenerate complex [[quadratic space]] of dimension {{mvar|n}}, and one associated orthogonal group, usually denoted {{math|O(''n'', '''C''')}}. It is the group of ''complex orthogonal matrices'', complex matrices whose product with their transpose is the identity matrix. As in the real case, {{math|O(''n'', '''C''')}} has two connected components. The component of the identity consists of all matrices of determinant {{math|1}} in {{math|O(''n'', '''C''')}}; it is denoted {{math|SO(''n'', '''C''')}}. The groups {{math|O(''n'', '''C''')}} and {{math|SO(''n'', '''C''')}} are complex Lie groups of dimension {{math|''n''(''n'' β 1) / 2}} over {{math|'''C'''}} (the dimension over {{math|'''R'''}} is twice that). For {{math|''n'' β₯ 2}}, these groups are noncompact. As in the real case, {{math|SO(''n'', '''C''')}} is not simply connected: For {{math|''n'' > 2}}, the [[fundamental group]] of {{math|SO(''n'', '''C''')}} is [[Cyclic group|cyclic of order 2]], whereas the fundamental group of {{math|SO(2, '''C''')}} is {{math|'''Z'''}}.
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