论文标题
有限的nilpotent组中两个元素的乘积顺序
The order of the product of two elements in finite nilpotent groups
论文作者
论文摘要
小组理论中的一个旧问题是描述元素在乘法下的行为方式。要概括一些经典的界限，以$ \ mathrm o（ab）$ a，两个元素的$ a，b $，一个有限的亚伯利亚组中的b $ to to nocuncutative案例，我们将$ \ mathrm o（ab）$替换为相互订单$ \ mathrm o（mathrm o（a，b）$，定义为protical integer $ n $ $ n $ n = n = n = n = n = n = n n = n = n = n = n = n n = n n = n = n = n = n = n n = n = n.然后，我们进行了比较$ \ MATHRM O（AB）$和$ \ MATHRM O（a，b）$中的nilpotent组中的$，并表明，在一组$γ$的组中，比率$ \ mathrm o（ab）/\ mathrm o（a，a，b）$（a，a，b）$ smitiont in Do $ sebs $ seled $ s sep ynaty e;超过$γ$。特别是，我们概括了P. Hall的结果，该结果断言$ \ mathrm o（ab）= \ mathrm o（a，b）$ in $ p $ groups，带有$ p>γ$。我们对2类组的组进行了更详细的分析，该组允许人们对$ \ Mathrm O（AB）/\ Mathrm O（a，b）$进行更明确的描述。
An old problem in group theory is that of describing how the order of an element behaves under multiplication. To generalize some classical bounds concerning the order $\mathrm o(ab)$ of two elements $a, b$ in a finite abelian group to the non-commutative case, we replace $\mathrm o(ab)$ with a notion of mutual order $\mathrm o(a, b)$, defined as the least positive integer $n$ such that $a^nb^n = 1$. Motivated by this, we then compare $\mathrm o(ab)$ and $\mathrm o(a, b)$ in finite nilpotent groups, and show that in a group of class $γ$, the ratio $\mathrm o(ab)/\mathrm o(a, b)$ lies in some fixed finite set $S(γ) \subset \mathbb Q$, whose elements do not involve prime factors exceeding $γ$. In particular, we generalize a result of P. Hall, which asserts that $\mathrm o(ab) = \mathrm o(a, b)$ in $p$-groups with $p > γ$. We end with a more detailed analysis for groups of class 2, which allows one to give a more explicit description of $\mathrm o(ab)/\mathrm o(a, b)$.
