论文标题
算术进展中的共轭类规模
Conjugacy class sizes in arithmetic progression
论文作者
论文摘要
令$ {\ rm cs}（g）$表示组$ g $的一组共轭类大小，然后$ {\ rm cs}^*（g）= {\ rm cs}（g）\ setMinus \ setMinus \ {1 \} $是非中央类别的尺寸。我们证明了三个结果。我们将所有有限组的$ g $与$ {\ rm cs}（g）= \ {a+d，\ dots，a+rd \} $ a算术前进2 $分类。 （我们表明$ {\ rm cs}（g）= \ {1,2,3 \} $。）我们最实质性的结果将所有$ g $与$ {\ rm cs}^*（g）= \ {2,4,4,6 \} $分类。最后，我们对所有最大的两个非中心共轭类规模的$ g $进行分类。 （在这里并不明显，但是$ {\ rm cs}^*（g）$的确有两个元素，因此算术进度也是如此。）
Let ${\rm cs}(G)$ denote the set of conjugacy class sizes of a group $G$, and let ${\rm cs}^*(G)={\rm cs}(G)\setminus\{1\}$ be the sizes of non-central classes. We prove three results. We classify all finite groups $G$ with ${\rm cs}(G)=\{a, a+d, \dots ,a+rd\}$ an arithmetic progression with $r\geqslant 2$. (We show that ${\rm cs}(G)=\{1,2,3\}$.) Our most substantial result classifies all $G$ with ${\rm cs}^*(G)=\{2,4,6\}$. Finally, we classify all groups $G$ whose largest two non-central conjugacy class sizes are coprime. (Here it is not obvious but it is true that ${\rm cs}^*(G)$ has two elements, and so is an arithmetic progression.)
