Editing Symplectic group (section)

{{Short description|Mathematical group}}{{for| finite groups with all characteristic abelian subgroups cyclic|group of symplectic type}}
{{Lie groups |Classical}}
{{Group theory sidebar |Topological}}
In [[mathematics]], the name '''symplectic group''' can refer to two different, but closely related, collections of mathematical [[Group (mathematics)|groups]], denoted {{math|Sp(2''n'', '''F''')}} and {{math|Sp(''n'')}} for positive integer ''n'' and [[field (mathematics)|field]] '''F''' (usually '''C''' or '''R'''). The latter is called the '''compact symplectic group''' and is also denoted by <math>\mathrm{U
Sp}(n)</math>. Many authors prefer slightly different notations, usually differing by factors of {{math|2}}. The notation used here is consistent with the size of the most common [[Matrix (math)|matrices]] which represent the groups. In [[Élie Cartan|Cartan]]'s classification of the [[simple Lie algebra]]s, the Lie algebra of the complex group {{math|Sp(2''n'', '''C''')}} is denoted {{math|''C<sub>n</sub>''}}, and {{math|Sp(''n'')}} is the [[Real form (Lie theory)#Compact real form|compact real form]] of {{math|Sp(2''n'', '''C''')}}. Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension {{math|''n''}}.

The name "[[Symplectic topology|symplectic]] group" was coined by [[Hermann Weyl]] as a replacement for the previous confusing names ('''line''') '''complex group''' and '''Abelian linear group''', and is the Greek analog of "complex".

The [[metaplectic group]] is a double cover of the symplectic group over '''R'''; it has analogues over other [[local field]]s, [[finite field]]s, and [[adele ring]]s.