For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for Z32, and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) [3]. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are Z32 and Z33 by a probabilistic approach in (1986) [11]. In this note, it is shown that every finite group of odd order has a TmSR for every m≥2.

中文翻译：

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关于有限群的锦标赛 m 半正则表示的说明


对于正整数 m，如果存在一个锦标赛 Γ使得 Γ 的自同构群与 G 同构，并且半正则作用在具有 m 个轨道的 Γ 的顶点集上，则称该组 G 接受一个锦标赛 m 半正则表示（简称 TmSR）。很容易看出，每个偶数阶的有限群都不允许任何正整数 m 的 TmSR。T1SR 是众所周知的锦标赛常规代表（简称 TRR）。在 1970 年代，Babai 和 Imrich 证明了除 Z32 之外的每个奇数阶有限群都允许 TRR，并且在 （1979） [3] 中，每个没有 2 阶元素的群（有限或无限）都允许 T2SR。后来，Godsil 在 （1986） [11] 中通过概率方法证明了唯一没有 TRR 的奇数阶有限群是 Z32 和 Z33，从而纠正了结果。在此注释中，显示了每个奇数阶的有限组对于每个 m≥2 都有一个 TmSR。