Nongenerator

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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 \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.