Does a Group being Finite Imply that It Is Cyclic?

Question

I have been studying Abstract Algebra, and all the finite groups that we have studied so far have also been cyclic. So, is it true that all finite groups are cyclic?
If yes, what is the theorem? If no, please provide a counterexample.

Answer 1

No. Try and find a generator for $$\mathbb{Z}_2 \times \mathbb{Z}_2$$ Shouldn't take long, there's only four possibilities. Next try:

$$\mathbb{Z}_6 \times \mathbb{Z}_2$$

In general, a finite group will be cyclic if and only if it is isomorphic to some direct product of cyclic groups with relatively prime orders. For example:

$$\mathbb{Z}_2 \times \mathbb{Z}_3, \ \ \text{ and } \ \ \mathbb{Z}_\mathbb{11} \times \mathbb{Z}_4$$

are cyclic. But of course, not all finite groups are Abelian (commutative) let alone cyclic.

Answer 2

For abelian take klein 4 group and for non abelian take dihedral group of order 8.