Problems(1)
Let (G,∗) a group of n>1 elements, and let g∈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∈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∈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×G→G that satisfy the following properties
(i) ∗ is asociative: ∀a,b,c∈G(a∗b)∗c=a∗(b∗c);
(ii) there exists an identity element e∈G such that ∀a∈G,a∗e=e∗a=a;
(iii) there exists inverse elements: ∀a∈G∃a−1∈G such that a∗a−1=a−1∗a=e. CIIM