Nongenerator

Symbol-free definition
An element in a group is said to be a nongenerator if it satisfies the following equivalent conditions:


 * Whenever a set containing it is a generating set for the group, the set obtained after removing the element is also a generating set. In other words, it is redundant in every generating set of the group.
 * It lies in every maximal subgroup
 * It lies inside the Frattini subgroup

Definition with symbols
An element $$g$$ in a group $$G$$ is termed a nongenerator if it satisfies the following equivalent conditions:


 * Whenever $$S$$ is a generating set for $$G$$ such that $$g \in S$$, $$S \setminus \{g\}$$ is also a generating set for $$G$$.
 * Whenever $$M \le G$$ is a maximal subgroup, $$g \in M$$
 * $$g \in \Phi(G)$$

Facts
The nongenerators form a group. This fact is not directly obvious but follows from the characterization of nongenerators as elements that lie inside the Frattini subgroup.