1-completed subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

Symbol-free definition

A subgroup of a group is said to be 1-completed if there is a single element outside the subgroup such that that element, along with the subgroup, generates the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be 1-completed if there is an element $x$ in $G$ such that the subgroup generated by $H$ and $x$ is the whole of $G$.

Relation with other properties

Stronger properties

• Maximal subgroup
• Cyclic-quotient subgroup

Weaker properties

• Finitely completed subgroup
• Abelian-completed subgroup

Metaproperties

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties