论文标题
诱导牙套和Hopf Galois结构
Inducing braces and Hopf Galois structures
论文作者
论文摘要
让$ p $为质量数字，让$ n $是$ p $无法排除的整数，这样每组订单$ np $都有正常的子组$ p $。 （尤其是$ p> n $。）我们证明，可以作为尺寸$ np $的左括号作为尺寸$ p $的唯一左括号和尺寸$ n $的左撑杆的半领产品获得。我们提供了一种方法，可以从尺寸$ n $的牙套中确定所有尺寸$ np $的牙套，以及从这些尺寸$ n $的乘法组到$ n $的乘法组到$ \ mathrm {z} _p^*$的某些类别的形态。从中，我们得出了一个公式，给出了Abelian Type $ \ Mathrm {Z} _P \ times E $的Hopf Galois结构数量，该公式在GALOIS $ np $ e $的Hopf Galois结构数量方面，$ np $ np $ np $ np $ np $。对于Prime Number $ p \ geq 7 $，我们应用获得的结果来描述所有尺寸$ 12p $的左括号，并确定Abelian Type的Hopf Galois结构数量，以$ 12p $的Galois扩展。
Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be obtained as a semidirect product of the unique left brace of size $p$ and a left brace of size $n$. We give a method to determine all braces of size $np$ from the braces of size $n$ and certain classes of morphisms from the multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it we derive a formula giving the number of Hopf Galois structures of abelian type $\mathrm{Z}_p \times E$ on a Galois extension of degree $np$ in terms of the number of Hopf Galois structures of abelian type $E$ on a Galois extension of degree $n$. For a prime number $p\geq 7$, we apply the obtained results to describe all left braces of size $12p$ and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree $12p$.
