Subgroup of index two

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 of index two if its index in the group is two, or equivalently, if it has exactly one coset other than itself.

Definition with symbols

A subgroup $N$ of a group $G$ is said to be of index two if $[G:N] = 2$.

Formalisms

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

The property of being a subgroup of index two can be expressed in first-order logic (in fact, the property of being a subgroup of any fixed finite index can be expressed in first-order logic).

Relation with other properties

Weaker properties

• Normal subgroup: For full proof, refer: index two implies normal
• Subgroup of prime index
• Abelian-quotient subgroup

External links

Links to related riders

• Mathlinks discussion forum page for Group with exactly n subgroups of index two