If G is finite group of order 3, then?

Question

a) G is always abelian
b) G is always non-abelian
c) G may be abelian
d) none

Answer 1

The answer is (a)

Suppose that the elements in G are {e , a , b } , where e denotes the identity in G .

Consider the element ab , we know that ab=a or b or e
( ab=a and ab=b are impossible cases .)
Thus ab=e and hence b=a^(-1)

G={e,a,a^(-1)} So G is abelian.

Answer 2

G is always abelian. In fact G is cyclic,
because any group of prime order is cyclic.

Answer 3

Not only is it abelian, it is cyclic.