Metaplectic group:Mp(2,R)

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Definition

This group can be defined in the following equivalent ways:

  1. It is the metaplectic group Mp(2,\R), i.e., it has degree two over the field of real numbers.
  2. 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 SL(2,\R) in Mp(2,\R), and then to the whole group. The successive quotients are cyclic group:Z2, cyclic group:Z2, and PSL(2,R). See also projective special linear group is simple.
chief length 3 The unique chief series is the same as the composition series above.
dimension of a real Lie group 3 As Mp(n,\R), n = 2: n(n+1)/2 = 2(2 + 1)/2 = 3
As double cover of SL(2,\R): Same as dimension of SL(2,\R), which is 3.

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