AI 中文总结
研究花环积中共轭问题的可解性条件。通过指出简·马修斯结论的不足，给出新的附加条件。还给出了置换受限花环积中共轭问题可解的充要条件，涉及群\(A\)、\(B\)的共轭问题、幂问题、阶问题等多种条件。
AI 中文摘要
1966年，简·马修斯声称，两个非平凡群\(A\)和\(B\)的标准受限花环积\(A \wr B\)中的共轭问题可解，当且仅当：(i)\(A\)和\(B\)中的共轭问题可解；(ii)\(B\)有一个可解的幂问题。我们表明，应该有一个附加条件，即要么\(A\)是阿贝尔群，要么\(B\)有一个可解的阶问题。我们还表明，如果\(A\)和\(B\)是非平凡递归表示的群，其中\(A\)有无限多个共轭类，并且\(B\)在\(B/H\)上可迁作用，那么置换受限花环积\(A \wr_{B/H} B\)中的共轭问题可解，当且仅当以下条件成立：(1)\(A\)和\(B\)中的共轭问题可解；(2)要么\(A\)是阿贝尔群，要么\(B\)中的轨道阶问题可解；(3)对于任何\(\gamma, \beta \in B\)，\(H \gamma \langle \beta \rangle\)的成员问题可解；(4)对于任何\(\beta \in B\)和任何有限的\(n\)对元素\((\alpha_i, \gamma_i)\)，其中\(\alpha_i, \gamma_i \in B\)，我们可以确定\(\big{[} \bigcap_{i=1}^n \alpha^{-1}_iH\gamma_i \langle \beta \rangle\big{]} \cap C_B(\beta) = \emptyset\)是否成立。
英文摘要
In 1966 Jane Matthews claimed that the conjugacy problem is solvable in the standard restricted wreath product $A \wr B$ of two nontrivial groups $A$ and $B$ if and only if (i) the conjugacy problem is solvable in $A$ and $B$ and (ii) $B$ has a \textit{solvable power problem}. We show that there should be an additional condition that either $A$ is abelian or $B$ has a \textit{solvable order problem}. We also show that, if $A$ and $B$ are non-trivial recursively presented groups where $A$ has an infinite number of conjugacy classes and $B$ acts on $B/H$ transitively, then the conjugacy problem in the permutational restricted wreath product $A \wr_{B/H} B$ is solvable if and only if the following hold: (1) the conjugacy problem is solvable in $A$ and in $B$; (2) either $A$ is abelian or \textit{the orbit order problem} is solvable in $B$; (3) for any $γ, β\in B$ the membership problem for $H γ\langle β\rangle$ is solvable; and (4) for any $β\in B$ and any finite set of $n$ pairs of elements $(α_i, γ_i)$, where $α_i, γ_i \in B$ we can determine whether or not $ \big{[} \bigcap_{i=1}^n α^{-1}_iHγ_i \langle β\rangle\big{]} \cap C_B(β) = \emptyset $.
