Transitive Avoidance Games on Boards of Odd Size

Avoidance (or misère) games are a type of positional games. Two players alternately claim points of a set $N$ (the `board' of the game). The game is determined by a family $L$ of subsets of $N$ and the following rule: The first player who claims every point of some set in $L$ loses the avoidance game. The game is called transitive if the group of all permutations of $N$ leaving $L$ invariant acts transitively on $N$. Johnson, Leader and Walters show that for a board size which is neither a prime number nor a power of two there are transitive avoidance games where the first player can force his win. For primes of size at least $17$, the corresponding problem remained open. We are going to close this gap and prove that for all primes $n$ of size at least $17$ there are also transitive avoidance games with board size $n$ where the first player can force his win.