A note on tournament m-semiregular representations of finite groups

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 $Z_3^2$, 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 $Z_3^2$ and $Z_3^3$ 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 $m\ge 2$.