# Nongenerator

From Groupprops

*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 in a group is termed a nongenerator if it satisfies the following equivalent conditions:

- Whenever is a generating set for such that , is also a generating set for .
- Whenever is a maximal subgroup,

## 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.