# Mathieu group:M11

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## Contents

## Definition

This group, termed the **Mathieu group of degree eleven** and denoted is the subgroup of the symmetric group of degree eleven defined by the following generating set:

.

Note that since both the generating permutations are even permutations, is in fact a subgroup of the alternating group of degree eleven.

This is one of the five simple Mathieu groups, which form a subset of the sporadic simple groups. The parameters for the simple Mathieu groups are . There are also Mathieu groups for parameters , but these are not sporadic simple groups. The Mathieu group for parameter is a simple group that is not a sporadic simple group, it is isomorphic to the projective special linear group:PSL(3,4).

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 7920 | groups with same order | |

exponent of a group | 1320 | groups with same order and exponent of a group | groups with same exponent of a group | |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length |

## Group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | No | |

nilpotent group | No | |

solvable group | No | |

simple group | Yes | |

minimal simple group | No |

## GAP implementation

### Definition using the Mathieu group function

The Mathieu group has order . Unfortunately, GAP does not assign group IDs for groups of such large orders. However, this group can be defined using the MathieuGroup function, as:

`MathieuGroup(11)`