Given a free group $\Gamma$ of finite rank $n$ and a prime number $p,$ denote by $\Gamma_k^\bullet$ the $k^\text{th}$ layer of the Stallings ($\bullet=S$) or Zassenhaus ($\bullet=Z$) $p$-central series, by $\mathcal{N}_{k}^\bullet$ the quotient $\Gamma/\Gamma_{k+1}^\bullet$ and by $\mathcal{L}_{k}^\bullet$ the quotient $\Gamma_k^\bullet /\Gamma_{k+1}^\bullet.$ In this paper we prove that there is a non-central extension of groups $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1, \mathcal{L}^\bullet_{k+1}) \longrightarrow Aut\;\mathcal{N}^\bullet_{k+1} \longrightarrow Aut \;\mathcal{N}^\bullet_k \longrightarrow 1, $ which splits if and only if $k=1$ and $p$ is odd if $\bullet=Z$ or, $k=1$ and $(p,n)= (3,2), (2,2)$ if $\bullet=S$. Moreover, if we denote by $IA^p(\mathcal{N}^\bullet_k )$ the subgroup of $Aut \;\mathcal{N}^\bullet_k$ formed by the automorphisms that acts trivially on $\mathcal{N}_1^\bullet,$ then the restriction of this extension to $IA^p(\mathcal{N}^\bullet_{k+1})$ give us a non-split central extension of groups $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1,\mathcal{L}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_k ) \longrightarrow 1. $

中文翻译：

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降 mod-p 中心级数的自同构

给定一个有限秩为 $n$ 的自由群 $\Gamma$ 和一个素数 $p，$ 用 $\Gamma_k^\bullet$ 表示 Stallings 的 $k^\text{th}$ 层（$\bullet= S$) 或 Zassenhaus ($\bullet=Z$) $p$-central 系列，由 $\mathcal{N}_{k}^\bullet$ 商 $\Gamma/\Gamma_{k+1}^\ bullet$ 并通过 $\mathcal{L}_{k}^\bullet$ 商$\Gamma_k^\bullet /\Gamma_{k+1}^\bullet.$ 在本文中我们证明存在一个非群的中心扩展 $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1, \mathcal{L}^\bullet_{k+1}) \longrightarrow Aut\;\mathcal{N}^\bullet_{k+1 } \longrightarrow Aut \;\mathcal{N}^\bullet_k \longrightarrow 1, $ 分裂当且仅当 $k=1$ 和 $p$ 是奇数如果 $\bullet=Z$ 或 $k=1$和 $(p,n)= (3,2), (2,2)$ 如果 $\bullet=S$。此外，如果我们用 $IA^p(\mathcal{N}^\bullet_k )$ 表示 $Aut \ 的子群；