# 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