自同构立方为大部分元素的有限群
Finite groups with an automorphism cubing a large fraction of elements
摘要：
我们调查可能的结构强加于一个有限群，其拥有的自同构发送一个大部分的组元素到他们的立方体，哲学是，这应该迫使该组，在某种意义上，接近阿贝尔。我们证明了两个定理。首先，我们完全分类所有有限群的自同构立方超过一半的元素。所有这样的群要么是幂零类2，要么有指数至多为2的阿贝尔子群。对于我们的第二个定理，我们证明了，如果一个群拥有一个自同构，将超过4/15的元素发送到它们的立方体，那么它一定是可解的。A_5组表明这一结果是最好的。我们的主要发现密切平行的结果，以前的作者对有限群拥有一个自同构，反转许多群元素。新的证明的技术性更微妙，也抛出了一个很好的连接到一个基本问题，在组合数论，即研究的子集有限循环群，避免非平凡的解决方案，一个或多个平移不变线性方程。
英文摘要：
We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or have an abelian subgroup of index at most 2. For our second theorem we show that, if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many group elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.
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