Originally Posted by

**biggybarks** Is this right.

**Suppose that G is an Abelian Group with an odd number of elements.?**

Show that the product of all of the elements of G is the identity

For ease, let's just show Abelian G with 7 elements. g = {e, a, b, c, d, f, g} --> e is the identity element.

Now consider a*b*c*d*e*f*g. Since the group is finite and abelian, then one of {b, c, d, f, g} must be the multiplicative inverse of a. Without loss of generality, we can presume that d = a^-1, f = b^-1 and g = c^-1.

a*b*c*d*e*f*g = a*d*b*f*e*c*g (commutative property)

= (a*d)*(b*f)*e*(c*g) (associative property)

= e*e*e*e (since d = a^-1, f = b^-1 and g = c^-1)

= e