3 Give an example of a nite group that is not cyclic or expl

3. Give an example of a nite group that is not cyclic or explain why no example
exists.

Solution

Klein V group is the simplest example which has an order of 4 and is isomorphic to Z2×Z2

Let G = {e,a,b,ab} where ba = ab, and a^2 = b^2 = e.

Every non-identity element is of order 2, so there is no element of order 4 which a cyclic group of order 4 must possess.