Klein four groups, cyclic groups

Question

A basic definition question:

"the Klein four group V is the simplest group that is not cyclic."

Does this simply mean you need two (non-identity) elements of the group to generate the entire group? (as opposed to only one element?)

Also, why is the distinction of cyclic vs non-cyclic important?

Answer 1

Groups of order $2$ and $3$ are cyclic. So the first possibility of having a non-cyclic group could be for order $4$ and Klein-4 happens to be one.

A non-cyclic group simply means it CANNOT be generated by just one element. In case of the Klein-4 group ($\cong \mathbb{Z}_2 \times \mathbb{Z}_2$), it can be generated by two elements ($(1,0), (0,1)$). However in general it doesn't mean that two elements will generate a non-cyclic group. For example take $(\mathbb{R},+)$, it is non-cyclic and cannot be generated by two elements (can you see why?).

The reason cyclic groups are sought after is because of their simplicity. Only one element (the generator) can tell you pretty much everything about the group.

Answer 2

Since it's not cyclic, it means it has no element of order $4$. While it's not generated by one element, it does turn out that it's generated by two, since the given group has the presentation $\langle a,b|a^2=b^2=e\rangle.$ This isn't really what it was trying to say, though. It meant that all smaller order groups are cyclic, making this the "simplest" example. Also, cyclic groups are important because they are extremely easy to understand; all the information of the group is contained in a single element. Groups with more generators are less easy to understand, less "simple." The more generators that you have, the harder it is to say meaningful things about your group.