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Putnam 1960 A7

Source: Putnam 1960

June 11, 2022
Putnamgroup theoryOrdersymmetric group

Problem Statement

Let N(n)N(n) denote the smallest positive integer NN such that xN=ex^N =e for every element xx of the symmetric group SnS_n, where ee denotes the identity permutation. Prove that if n>1,n>1, N(n)N(n−1)={p  if  n  is a power of a prime p1  otherwise.\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\ 1\; \text{otherwise}. \end{cases}
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