p-adic estimates of the number of permutation representations

Abstract

Let p be a prime, and let G be a finite abelian p-group of type $\lambda =({\lambda }_1,{\lambda }_2,\dots )$ with ${\lambda }_1\ge {\lambda }_2\ge \cdots$ and $\sum {\lambda }_i=s$. Set $u=\max \{{\lambda }_1,[(s+1)/2]\}$ and $v=s-u$. For each nonnegative integer n, let $h_n(G)$ be the number of homomorphisms from G to the symmetric group $S_n$ on n letters. Except for the case where $p=2$ and $u+{\delta }_{v0}\le v+1$, δ the Kronecker delta, or $p=3$ and $u=v\ge 1$, there exist p-adic analytic functions $f_r(X)$ for $r=0,1,\dots ,p^{u+1}-1$ and a polynomial $\eta (X)$ with integer coefficients such that for any nonnegative integer y, $h_{p^{u+1}y+r}(G)=p^{\{{\sum }_{j=1}^u{p}^j-(u-v)\}y}{f}_r(y){\prod }_{j=1}^y\eta (j)$ and ${\mathrm{ord}}_p(h_{p^{u+1}y+r}(G))=\{{\sum }_{j=1}^u{p}^j-(u-v)\}y+{\mathrm{ord}}_p(f_r(y))$. If $p=2$, ${\lambda }_3=0$, and $u=v\ge 1$ or if $p=3$ and $u=v\ge 1$, then $h_n(G)$ has analogous properties. Under the assumption that ${\lambda }_3=0$, some results for the number of permutation representations of G in the wreath product of a cyclic group of order p with $S_n$ are also presented.