CIIM 2019 Problem 4

Let $(G, *)$ a group of $n > 1$ elements, and let $g \in G$ be an element distinct from the identity. Ana and Bob play with the group $G$ on the following way: Starting with Ana and playing alternately, each player selects an element of $G$ that has not been selected before, until each element of $G$ have been selected or a player have selected the elements $a$ and $a * g$ for some $a \in G$ . In that case it is said that the player loses and his opponent wins. $a)$ If $n$ is odd, show that, independent of element $g$ , one of the two players has a winning strategy and determines which player possesses such a strategy. $b)$ If $n$ is even, show that there exists an element $g \in G$ for which none of the players has a winning strategy.

Note: A group $(G, *)$ es a set $G$ together with a binary operation $* : G\times G \to G$ that satisfy the following properties $(i)$ $*$ is asociative: $\forall a, b, c \in G (a * b) * c = a * (b * c)$ ; $(ii)$ there exists an identity element $e \in G$ such that $\forall a \in G, a *e = e * a = a;$ $(iii)$ there exists inverse elements: $\forall a \in G \exists a^{-1} \in G$ such that $a*a^{-1} = a^{-1} *a = e.$