AI 中文总结
该研究确定了Aut(Fₙ)的下三角子群Dₙ和Aₙ⁺的最小基数生成集及换位子形式的定义关系，丰富了自由群自同构群子群表示的相关结果。
AI 中文摘要
设Fₙ为秩n≥2的自由群，基为x₁,…,xₙ。对1≤j<i≤n，令d_{i,j}、e_{i,j}为Fₙ的自同构，满足d_{i,j}(x_i)=x_i x_j、e_{i,j}(x_i)=x_j x_i，且固定其余自由生成元。记Dₙ=⟨d_{i,j}|1≤j<i≤n⟩，Aₙ⁺=⟨d_{i,j},e_{i,j}|1≤j<i≤n⟩。我们证明Dₙ在生成元d_{r+1,r}（1≤r≤n-1）上有表示，含权为2、3、4的三类换位子关系；将其推广到Aₙ⁺在生成元d_{r+1,r}、e_{r+1,r}（1≤r≤n-1）上的表示，这些生成集基数最小，且Aₙ⁺的表示可选每个定义关系为单个换位子。
英文摘要
Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.
