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

From Groupprops
Jump to: navigation, search
Line 5: Line 5:
 
The '''order statistics''' of a [[finite group]] is a function <math>\mathbb{N}_0 \to \mathbb{N}_0</math> which takes <math>n</math> and outputs the number of elements <math>x</math> whose order is <math>n</math>.
 
The '''order statistics''' of a [[finite group]] is a function <math>\mathbb{N}_0 \to \mathbb{N}_0</math> which takes <math>n</math> and outputs the number of elements <math>x</math> whose order is <math>n</math>.
  
If <math>f</math> denotes the order statistics function, then the Dirichlet convolution <math>F = f * U</math> gives, for each <math>n</math>, the number of elements <math>x</math> satisfying <math>x^n = e</math>.
+
If <math>f</math> denotes the order statistics function, then the [[number:Dirichlet convolution|Dirichlet convolution]] <math>F = f * U</math> gives, for each <math>n</math>, the number of elements <math>x</math> satisfying <math>x^n = e</math>.
 +
 
 +
Two finite groups that have the same order statistics are termed [[order statistics-equivalent finite groups]].
  
 
==Facts==
 
==Facts==

Revision as of 22:26, 24 June 2009

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

Definition

The order statistics of a finite group is a function \mathbb{N}_0 \to \mathbb{N}_0 which takes n and outputs the number of elements x whose order is n.

If f denotes the order statistics function, then the Dirichlet convolution F = f * U gives, for each n, the number of elements x satisfying x^n = e.

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 n, the number F(n) of n^{th} roots of the identity is a multiple of the gcd of n and the order of the group.

Number of elements of prime order is nonzero

Further information: Cauchy's theorem

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

Other facts