Orbit sizes under automorphism actions in finite groups
Orbit sizes under automorphism actions in finite groups
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DOI：
10.1016/0021-8693(83)90155-2
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发表时间：
1983-06
影响因子：
0.9
通讯作者：
T. Yuster
中科院分区：
文献类型：
--
作者：
T. Yuster
研究了有限群H通过自同构作用在有限群G上的情形。这一情况的一个方面没有得到太多讨论，这就是以下问题：在这种行动下可能出现哪些轨道大小？答案当然取决于G和H的结构。本文并没有完全回答这个问题，而是通过考虑一个相关的问题，发现了在一定条件下G和H上的这组轨道大小必须满足的一些条件。如果H/K是幂零的，则定理1.5断言，如果x EG在大小为m的H轨道上，y EG在大小为n的H轨道上，且（m，n）= 1，则xy在大小为mn的H轨道上。这是本文的主要结果。我们看到，在G和H上的这些条件下，这个结果排除了包含整数nz和n且（m，n）= 1的轨道大小的集合，除非该集合还包含整数mFi。
The situation when a finite group H acts by automorphisms on a finite group G has been studied in some detail. One aspect of this situation which has not been much discussed is the following question: What possible sets of orbit sizes can arise under such actions? Certainly the answer will depend on the structures of G and H. In this paper we do not answer this question completely but, by considering a related problem, discover some conditions which must be met by that set of orbit sizes under certain conditions on G and H.Suppose that 7~ is the set of primes involved in (GI and K= O,,(H). If H/K is nilpotent, then Theorem 1.5 asserts that if x EG is in an H orbit of size m and y EG is in an H orbit of size n with (m, n)= 1, then xy is in an H orbit of size mn. This is the main result of the paper. We see that under these conditions on G and H, this result precludes a set of orbit sizes containing integers nz and n with (m, n)= 1 unless the set also includes the integer mFi.