MathDB

Let

N(n)

denote the smallest positive integer

N

such that

x^N =e

for every element

x

of the symmetric group

S_n

, where

e

denotes the identity permutation. Prove that if

n>1,

\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}

A7