# Difference between revisions of "Order statistics of a finite group"

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==Definition== | ==Definition== | ||

− | + | ===Individual version=== | |

− | If <math>f</math> denotes the order statistics function, then the Dirichlet convolution <math>F = f * U</math> gives, for each <math> | + | The '''order statistics''' of a [[finite group]] is a function <math>\mathbb{N}_0 \to \mathbb{N}_0</math> which takes <math>d</math> and outputs the number of elements <math>x</math> whose order is <math>d</math>. The function is usually restricted only to divisors of the order of the group, because [[order of element divides order of group]]. |

+ | |||

+ | ===Cumulative version=== | ||

+ | |||

+ | If <math>f</math> denotes the order statistics function, then the [[number:Dirichlet convolution|Dirichlet convolution]] <math>F = f * U</math> gives, for each divisor <math>d</math> of the order of the group, the number of elements <math>x</math> satisfying <math>x^d = e</math>. This function is termed the '''cumulative order statistics''' function. The order statistics can be deduced from the cumulative order statistics and vice versa. | ||

+ | |||

+ | Two finite groups that have the same order statistics are termed [[order statistics-equivalent finite groups]]. | ||

==Facts== | ==Facts== | ||

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===Number of nth roots is a multiple of n=== | ===Number of nth roots is a multiple of n=== | ||

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+ | {{further|[[Number of nth roots is a multiple of n]]}} | ||

For any <math>n</math>, the number <math>F(n)</math> of <math>n^{th}</math> roots of the identity is a multiple of the gcd of <math>n</math> and the order of the group. | For any <math>n</math>, the number <math>F(n)</math> of <math>n^{th}</math> roots of the identity is a multiple of the gcd of <math>n</math> and the order of the group. | ||

===Number of elements of prime order is nonzero=== | ===Number of elements of prime order is nonzero=== | ||

+ | |||

+ | {{further|[[Cauchy's theorem]]}} | ||

For any prime <math>p</math> dividing the order of the group, there is a cyclic subgroup of order <math>p</math>. Hence, <math>f(p) \ge p-1</math>. | For any prime <math>p</math> dividing the order of the group, there is a cyclic subgroup of order <math>p</math>. Hence, <math>f(p) \ge p-1</math>. | ||

+ | |||

+ | ==Relation with arithmetic functions== | ||

+ | |||

+ | The order statistics of a finite group can be used to deduce the values of the following arithmetic function: | ||

+ | |||

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

+ | ! Arithmetic function !! Meaning !! How it can be deduced from the order statistics | ||

+ | |- | ||

+ | | [[order of a group|order]] || the number of elements in the group || add up the number of elements of each order | ||

+ | |- | ||

+ | | [[exponent of a group|exponent]] || the lcm of the orders of all elements || take the lcm of all elements that give nonzero output under the order statistics function. | ||

+ | |} | ||

+ | |||

+ | ==Relation with group properties== | ||

+ | |||

+ | A better tabulated version of this information is available at [[order statistics-equivalent finite groups#Relation with group properties]]. | ||

+ | |||

+ | * [[Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring]] | ||

+ | * [[Order statistics of a finite group determine whether it is nilpotent]] | ||

+ | * [[Finite abelian groups with the same order statistics are isomorphic]] | ||

+ | |||

+ | ==GAP implementation== | ||

+ | |||

+ | GAP code can be written to output the order statistics of any finite group in a number of convenient formats. See the code for some such functions at [[GAP:OrderStatistics]] and [[GAP:CumulativeOrderStatistics]]. |

## Latest revision as of 13:35, 21 June 2010

*This article talks about a statistics, which could be a function or a set of numbers, associated with any finite group*

## Contents

## Definition

### Individual version

The **order statistics** of a finite group is a function which takes and outputs the number of elements whose order is . The function is usually restricted only to divisors of the order of the group, because order of element divides order of group.

### Cumulative version

If denotes the order statistics function, then the Dirichlet convolution gives, for each divisor of the order of the group, the number of elements satisfying . This function is termed the **cumulative order statistics** function. The order statistics can be deduced from the cumulative order statistics and vice versa.

Two finite groups that have the same order statistics are termed order statistics-equivalent finite groups.

## Facts

The order statistics function for a group cannot be chosen arbitrarily. It is subject to some constraints.

### Number of nth roots is a multiple of n

`Further information: Number of nth roots is a multiple of n`

For any , the number of roots of the identity is a multiple of the gcd of and the order of the group.

### Number of elements of prime order is nonzero

`Further information: Cauchy's theorem`

For any prime dividing the order of the group, there is a cyclic subgroup of order . Hence, .

## Relation with arithmetic functions

The order statistics of a finite group can be used to deduce the values of the following arithmetic function:

Arithmetic function | Meaning | How it can be deduced from the order statistics |
---|---|---|

order | the number of elements in the group | add up the number of elements of each order |

exponent | the lcm of the orders of all elements | take the lcm of all elements that give nonzero output under the order statistics function. |

## Relation with group properties

A better tabulated version of this information is available at order statistics-equivalent finite groups#Relation with group properties.

- Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring
- Order statistics of a finite group determine whether it is nilpotent
- Finite abelian groups with the same order statistics are isomorphic

## GAP implementation

GAP code can be written to output the order statistics of any finite group in a number of convenient formats. See the code for some such functions at GAP:OrderStatistics and GAP:CumulativeOrderStatistics.