Is it only the generator of the group that commutes with all the other elements?

Question

If a group is generated by an element does that mean the generator commutes with all the other elements or does it mean that because the group is cyclic(as it has a generator) that all elements commute with each other.

For example, I am trying to find the conjugacy classes of the group D4 and am not sure if I could use the property that elements commute with each other. It seems to be taking too long so I was wondering what would be some facts I could be using?

From the notation below I understand that D4 is generated by a and b. So, is it only these two elements that commute with the others? $$ D_4=\langle a, b\rangle=\{e, a, a^2, a^3, b, ab, a^2b, a^3b\} $$

Answer 1

I'm not sure if this answers your question, but here it goes.

If a group is generated by a single element $a$, then $a$ commutes with all other elements. In fact, the group is Abelian then.

But if a group is generated by more than one element, there is no reason to assume that the generators commute with all other elements; this happens if and only if the group is Abelian.

In particular, in $D_4$ it is not true that $a$ and $b$ commute with all other elements.

Answer 2

The short answer is: the idea of "generators" and "commutativity" are completely disjoint. I don't really know what else to say...

In $D_4$, the generators $a$ and $b$ do not commutate with each other, so cannot each commute with every element. For example, $ba=a^3b\neq ab$.