Symmetric Group

The symmetric group $S_n$ of Degree $n$ is the Group of all Permutations on $n$ symbols. $S_n$ is therefore of Order $n!$ and contains as Subgroups every Group of Order $n$. The number of Conjugacy Classes of $S_n$ is given by the Partition Function P.

Netto's Conjecture states that the probability that two elements $P_1$ and $P_2$ of a symmetric group generate the entire group tends to 3/4 as $n\to\infty$. This was proven by Dixon in 1967.

See also Alternating Group, Conjugacy Class, Finite Group, Netto's Conjecture, Partition Function P, Simple Group

References

Lomont, J. S. ``Symmetric Groups.'' Ch. 7 in Applications of Finite Groups. New York: Dover, pp. 258-273, 1987.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/html/contents.html#alt.