Abstract Let cs ⁢ ( G ) {\mathrm{cs}(G)} denote the set of conjugacy class sizes of a group G, and let cs * ⁢ ( G ) = cs ⁢ ( G ) ∖ { 1 } \mathrm{cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\} be the sizes of non-central classes. We prove three results. We classify all finite groups for which (1) cs ⁢ ( G ) = { a , a + d , … , a + r ⁢ d } {\mathrm{cs}(G)=\{a,a+d,\dots,a+rd\}} is an arithmetic progression with r ⩾ 2 {r\geqslant 2} ; (2) cs * ⁢ ( G ) = { 2 , 4 , 6 } {\mathrm{cs}^{*}(G)=\{2,4,6\}} is the smallest case where cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} are coprime. For (3), it is not obvious, but it is true that cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} has two elements, and so is an arithmetic progression.

中文翻译：

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等差数列中的共轭类大小

摘要 让 cs ⁢ ( G ) {\mathrm{cs}(G)} 表示组 G 的共轭类大小的集合，让 cs * ⁢ ( G ) = cs ⁢ ( G ) ∖ { 1 } \mathrm{ cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\} 是非中心类的大小。我们证明了三个结果。我们对所有有限群进行分类，其中 (1) cs ⁢ ( G ) = { a , a + d , … , a + r ⁢ d } {\mathrm{cs}(G)=\{a,a+d,\ dots,a+rd\}} 是一个等差数列，其中 r ⩾ 2 {r\geqslant 2} ；(2) cs * ⁢ ( G ) = { 2 , 4 , 6 } {\mathrm{cs}^{*}(G)=\{2,4,6\}} 是 cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} 是长度大于 2 的等差数列（我们最实质性的结果）；(3) cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} 的最大两个成员是互质的。对于（3），不明显，但确实 cs * ⁢ ( G ) {\mathrm{cs}^{*}(G)} 有两个元素，所以是一个等差数列。