\(p\)-divisibility of the number of solutions of \(x^p=1\) in a symmetric group
Let \(p\) be a prime number and let \(a(n)\) be the number of elements \(x\) of the symmetric group \(S_n\) that satisfy the equation \(x^p=1\). Let \(\text{ord}_p(a(n))\) denote the exponent of \(p\) in the decomposition of \(a(n)\) into prime factors. Put \(a(0)=1\) and define \( \gamma(n)=[\frac{n}{p}]-[\frac{n}{p^2}]\) for \( n=0,1,2,\cdots,\) where \([n]\) denotes the largest integer not exceeding a real number \(x\). In the paper under review, the authors give a lower bound for \(\text{ord}_p(a(n))\), in fact, they prove that \(\text{ord}_p(a(n)) \geq \gamma(n)\) and prove some theorems of which the following Theorem C is the main one: Theorem C. Let \(m\) be an arbitrary nonnegative integer and let \(r\) be an integer with \(p\leq r \leq p^2-1\). Assume that \(\text{ord}_p(a(r))=\gamma(r)+2\). Then \[ a(mp^2+r)\equiv (-1)^mp^{m(p-1)} \biggl(a(r)-a(r)m-\frac{a(p^2+r)}{p^{p-1}}m\biggr)\bmod p^{\gamma(mp^2+r)+3}. \] If \(a(p^2+r)\equiv -p^{p-1}a(r)\bmod p^{\gamma(p^2+r)+3}\), then \(\text{ord}_p(a(mp^2+r))=\gamma(mp^2+r)+2\); otherwise, there exists a \(p\)-adic integer \(b\) such that \(\text{ord}_p(a(mp^2+r))=\gamma(mp^2+r)+2+ \text{ord}_p(m-b)\). Theorem C has been generalized by \textit{K. Conrad} in an as yet unpublished preprint [\(p\)-adic properties of truncated Artin-Hasse coefficients].
scientific article; zbMATH DE number 1756465
scientific article