Metaplectic group:Mp(2,R)
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Definition
This group can be defined in the following equivalent ways:
- It is the metaplectic group
, i.e., it has degree two over the field of real numbers.
- It is the unique double cover of special linear group:SL(2,R).
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order of a group | cardinality of the continuum | Same infinite cardinality as SL(2,R). | |
exponent of a group | infinite | Same as SL(2,R). | |
composition length | 3 | We can construct a composition series that begins with a subgroup of order two for which the quotient is SL(2,R), then proceeds to the inverse image of the center of ![]() ![]() | |
chief length | 3 | The unique chief series is the same as the composition series above. | |
dimension of a real Lie group | 3 | As ![]() As double cover of ![]() ![]() |
Group properties
Property | Satisfied? | Explanation |
---|---|---|
abelian group | No | |
nilpotent group | No | |
solvable group | No | |
quasisimple group | Yes | The center is isomorphic to cyclic group:Z4 and the inner automorphism group is isomorphic to PSL(2,R). |
simple group | No | The center is proper and nontrivial |