Understanding the definition of "Frattini-free group"

Question

Is it true that in " Frattini-free group " all elements are generators other than identity.

https://groupprops.subwiki.org/wiki/Frattini_subgroup

I have difficulty in understanding the second part ie " It is the set of all nongenerators "

Answer 1

A non-generator is an element $x \in G$ satisfying the following property:

If $S$ is any set of generators of the group $G$, then so is $S \setminus \{ x\}$.

This means that $x$ does not appear in any minimal set of generators, in other words "to generate $G$, you don't need $x$".

In this sense, being a non-generator is not equivalent to the statement $\langle x \rangle \neq G$ ($x$ does not generate $G$).

For example, if you consider $G= C_2 \times C_2$, then its Fratini subgroup is trivial, however $G$ is not cyclic (for all $x \in G$ you have $\langle x \rangle \neq G$).