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

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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===
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{{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>.
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===Other facts===
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* [[Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring]]
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* [[Order statistics of a finite group determine whether it is nilpotent]]
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* [[Finite abelian groups with the same order statistics are isomorphic]]

Revision as of 20:56, 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.

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