# Subgroup of index two

From Groupprops

(Redirected from Normal 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]

## Contents

## 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 of a group is said to be **of index two** if .

## 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