# Difference between revisions of "Monster group"

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This group, termed the '''Monster''' or '''Monster group''' (and denoted by the shorthand <math>M</math>), is the largest sporadic simple group. It has order: | This group, termed the '''Monster''' or '''Monster group''' (and denoted by the shorthand <math>M</math>), is the largest sporadic simple group. It has order: | ||

− | <math>808017424794512875886459904961710757005754368000000000 = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71</math>. | + | <math>808017424794512875886459904961710757005754368000000000 = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71</math>. |

The prime divisors of the order of the monster group are precisely the same as the [[supersingular prime]]s. | The prime divisors of the order of the monster group are precisely the same as the [[supersingular prime]]s. |

## Latest revision as of 02:05, 24 November 2017

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]

## Contents

## Definition

This group, termed the **Monster** or **Monster group** (and denoted by the shorthand ), is the largest sporadic simple group. It has order:

.

The prime divisors of the order of the monster group are precisely the same as the supersingular primes.

The monster group is a sporadic simple group, and is the sporadic simple group of largest order. A sporadic simple group that is not isomorphic to a subgroup of the monster group is termed a pariah.

## Arithmetic functions

### Basic arithmetic functions

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 808017424794512875886459904961710757005754368000000000 | groups with same order |

### Arithmetic functions of a counting nature

Function | Value | Similar groups | Explanation for function value |
---|---|---|---|

number of conjugacy classes | 194 | groups with same order and number of conjugacy classes | groups with same number of conjugacy classes | See element structure of monster group, linear representation theory of monster group |

## GAP implementation

This group is too large to be stored and actively manipulated in GAP. However, some information about the group is stored in GAP under the symbol "M" -- for more, see linear representation theory of monster group.