论文标题
$ g $的花圈产品中的功率与$ s_n $
Powers in the wreath product of $G$ with $S_n$
论文作者
论文摘要
在本文中，我们计算花圈产品中的功率$ g \ wr s_n $，对于任何有限的组$ g $。对于$ r \ geq 2 $，p prime，请考虑$ω_r：g \ wr s_n \ to g \ wr s_n $由$ g \ mapsto g^r $定义。令$ p_ {r}（g \ wr s_n）= \ frac {|ω_r（g \ wr s_n）|} {| g |^n n！} $，这是$ g \ wr s_n $ is a $ r^{th th} $ power中随机选择的元素的可能性。我们证明，$ p_r（g \ wr s_ {n+1}）= p_r（g \ wr s_n）$ for All $ n \ not \ equiv -1（\ equiv -1（\ text {mod} r）$，如果$ g $的订单为$ g $。我们还为$ g \ wr s_n $中的$ r^{th} $ powers的共轭类数量提供了一个公式。
In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\geq 2$, a prime, consider $ω_r: G\wr S_n\to G\wr S_n$ defined by $g \mapsto g^r$. Let $P_{r}(G\wr S_n)=\frac{|ω_r(G\wr S_n)|}{|G|^n n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\not \equiv -1(\text{mod } r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.
