# Metaplectic group:Mp(2,R)

From Groupprops

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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 in , 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 As double cover of : Same as dimension of , 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 |