论文标题
算术进展中对共轭类规模的限制
Restrictions on sets of conjugacy class sizes in arithmetic progressions
论文作者
论文摘要
我们继续进行研究，该研究始于[3]和[4]，这些调查是有限的群体，其一组非平凡的共轭类规模构成了算术进展。令$ g $为有限组，并表示$ g $ $ {\ rm cs}（g）$的共轭类规模的集合。满足$ {\ rm cs}（g）= \ {1,2,4,6 \} $和$ \ {1,2,4,6,8 \} $的有限组分别在[4]和[3]中分类，我们分别通过以下来证明这些示例是特殊的。存在一个有限的$ g $，使得$ {\ rm cs}（g）= \ {1，2^α，2^{α+1}，2^α3\} $，仅当$α= 1 $时。此外，存在一个有限的$ g $，使得$ {\ rm cs}（g）= \ {1，2^α，2^{α+1}，2^α3，2^α3，2^{α+2}} \ \} $ and $α$和$α$是奇数If y If $ nif $α= 1 $。
We continue the investigation, that began in [3] and [4], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1,2,4,6\}$ and $\{1,2,4,6,8\}$ are classified in [4] and [3], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^α, 2^{α+1}, 2^α3 \}$ if and only if $α=1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^α, 2^{α+1}, 2^α3, 2^{α+2}\}$ and $α$ is odd if and only if $α=1$.
