Abstract The twin group T n {T_{n}} is a right-angled Coxeter group generated by n - 1 {n-1} involutions, and the pure twin group PT n {\mathrm{PT}_{n}} is the kernel of the natural surjection from T n {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in T n {T_{n}} , which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in T n {T_{n}} . We give a new proof of the structure of Aut ⁡ ( T n ) {\operatorname{Aut}(T_{n})} for n ≥ 3 {n\geq 3} , and show that T n {T_{n}} is isomorphic to a subgroup of Aut ⁡ ( PT n ) {\operatorname{Aut}(\mathrm{PT}_{n})} for n ≥ 4 {n\geq 4} . Finally, we construct a representation of T n {T_{n}} to Aut ⁡ ( F n ) {\operatorname{Aut}(F_{n})} for n ≥ 2 {n\geq 2} .

中文翻译：

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孪生群的共轭类和自同构

摘要 孪生群 T n {T_{n}} 是由 n - 1 {n-1} 次对合生成的直角 Coxeter 群，纯孪生群 PT n {\mathrm{PT}_{n}} 是从 T n {T_{n}} 到 n 个符号上的对称群的自然投影的核。在本文中，我们调查了这些群体的一些结构方面。我们推导出 T n {T_{n}} 中对合共轭类数的公式，有趣的是，它与著名的斐波那契数列有关。我们还推导出了 T n {T_{n}} 中对合 z 类数量的递归公式。我们给出了 Aut ⁡ ( T n ) {\operatorname{Aut}(T_{n})} 对于 n ≥ 3 {n\geq 3} 的结构的新证明，并证明 T n {T_{n}}同构于 Aut ⁡ ( PT n ) {\operatorname{Aut}(\mathrm{PT}_{n})} 的子群，对于 n ≥ 4 {n\geq 4} 。最后，