论文标题
具有许多有限顺序元素的小组
Profinite groups with many elements of bounded order
论文作者
论文摘要
Lévai和Pyber提出以下猜想： 让$ g $是一个概述的组，使得方程的解决方案$ x^n = 1 $具有正haar量度。然后，$ g $具有开放子组$ h $和一个元素$ t $，使得coset $ th $的所有要素都有订单划分$ n $（请参阅[kourovka Notebook，2019年第19号]的问题14.53）。 \\我们为所有有限组定义一个常数$ C_N $，并证明后者的猜想是等效于猜想的，说$ c_n <1 $。使用后一个等效性，我们观察到，莱维和派伯的猜想的正确性意味着在普遍的Hughes-Thompson子组$ H_N $ H_N $ H_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n $ h_n n n n n'notrivial nontrivial notrivial notrivial no-Trivial nongial的索引的存在。众所周知，即使对于所有Primes $ n = p \ geq 5 $，后者也广泛开放。对于Odd $ n $，我们还证明Lévai和Pyber的猜想相同表明，只有在有限的可解决方案组中计算出$ c_n $，$ c_n $小于$ 1 $。 \\猜想的有效性已在[Arch。数学。 （巴塞尔）75（2000）1-7]对于$ n = 2 $。在这里，我们确认$ n = 3 $的猜想。
Lévai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all elements of the coset $tH$ have order dividing $n$ (see Problem 14.53 of [The Kourovka Notebook, No. 19, 2019]). \\ We define a constant $c_n$ for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying $c_n<1$. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the universal upper bound $\frac{1}{1-c_n}$ on the index of generalized Hughes-Thompson subgroup $H_n$ of finite groups whenever it is non-trivial. It is known that the latter is widely open even for all primes $n=p\geq 5$. For odd $n$ we also prove that Lévai and Pyber conjecture is equivalent to show that $c_n$ is less than $1$ whenever $c_n$ is only computed on finite solvable groups. \\ The validity of the conjecture has been proved in [Arch. Math. (Basel) 75 (2000) 1-7] for $n=2$. Here we confirm the conjecture for $n=3$.
