MathDB

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)

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)

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.

Problem Statement