Dixon's Theorem (Group Theory)

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Theorem

Let $P_1$ and $P_2$ be distinct elements of the symmetric group on $n$ letters.


The probability that $\set {P_1, P_2}$ forms a generator of $S_n$ approaches $\dfrac 3 4$ as $n$ tends to infinity.


Proof


Source of Name

This entry was named for John D. Dixon.


Historical Note

Dixon's Theorem started out as a conjecture made by Eugen Otto Erwin Netto, and published by him in his $1882$ work Substitutionentheorie und ihre Anwendung auf die Algebra.

As a consequence, it was referred to as Netto's Conjecture.

It was finally proved by John D. Dixon in $1967$.

Since then it has been called Dixon's Theorem.


Sources