Nongenerator

This article defines a property of elements in groups

Definition

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 $M\leq G$ 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.