Metaplectic group:Mp(2,R)

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group can be defined in the following equivalent ways:

1. It is the metaplectic group $Mp(2,\mathbb {R} )$, 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 $SL(2,\mathbb {R} )$ in $Mp(2,\mathbb {R} )$, 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 $Mp(n,\mathbb {R} ),n=2:n(n+1)/2=2(2+1)/2=3$ As double cover of $SL(2,\mathbb {R} )$: Same as dimension of $SL(2,\mathbb {R} )$, 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