Dixon's Theorem (Group Theory)

This proof is about Dixon's Theorem in the context of Group Theory. For other uses, see Dixon's Theorem.

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

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Source of Name

This entry was named for John Douglas 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 Douglas Dixon in $1967$.

Since then it has been called Dixon's Theorem.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,75$