# Difference between revisions of "Metaplectic group:Mp(2,R)"

From Groupprops

(Created page with "{{particular group}} ==Definition== This group can be defined in the following equivalent ways: # It is the metaplectic group <math>Mp(2,\R)</math>, i.e., it has degree...") |
|||

(One intermediate revision by the same user not shown) | |||

Line 7: | Line 7: | ||

# It is the [[metaplectic group]] <math>Mp(2,\R)</math>, i.e., it has degree two over the [[field of real numbers]]. | # It is the [[metaplectic group]] <math>Mp(2,\R)</math>, 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)]]. | # It is the unique double cover of [[special linear group:SL(2,R)]]. | ||

+ | |||

+ | ==Arithmetic functions== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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)]]. | ||

+ | |- | ||

+ | | {{arithmetic function value|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 <math>SL(2,\R)</math> in <math>Mp(2,\R)</math>, 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]]. | ||

+ | |- | ||

+ | | {{arithmetic function value|chief length|3}} || || The unique chief series is the same as the composition series above. | ||

+ | |- | ||

+ | | {{arithmetic function value|dimension of a real Lie group|3}} || || As <math>Mp(n,\R), n = 2: n(n+1)/2 = 2(2 + 1)/2 = 3</math><br>As double cover of <math>SL(2,\R)</math>: Same as dimension of <math>SL(2,\R)</math>, which is 3. | ||

+ | |} | ||

+ | |||

+ | ==Group properties== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Property !! Satisfied? !! Explanation | ||

+ | |- | ||

+ | | [[dissatisfies property::abelian group]] || No || | ||

+ | |- | ||

+ | | [[dissatisfies property::nilpotent group]] || No || | ||

+ | |- | ||

+ | | [[dissatisfies property::solvable group]] || No || | ||

+ | |- | ||

+ | | [[satisfies property::quasisimple group]] || Yes || The center is isomorphic to [[cyclic group:Z4]] and the [[inner automorphism group]] is isomorphic to [[PSL(2,R)]]. | ||

+ | |- | ||

+ | | [[dissatisfies property::simple group]] || No || The center is proper and nontrivial | ||

+ | |} |

## Latest revision as of 16:23, 18 September 2012

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## 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 |