# Tour:Subgroup

**This article adapts material from the main article:** Subgroup

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Abelian group|UP: Introduction one (beginners)|NEXT: Trivial groupExpected time for this page: 10 minutes

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PREREQUISITES: Definition of group. Return to group if you do not remember this.WHAT YOU NEED TO DO:

- Understand thoroughly, the
definition in terms of closure of binary operationand theuniversal algebraic definitionof subgroup- Read, and make sense, of the other definitions
- Make sense of the examples given and use these to get some intuition about what being a
subgroupmeans

WHAT YOU DO NOT NEED TO DO: Prove that all the definitions are equivalent. This will be covered in part two of the tour.

IF YOU ARE FOLLOWING ANOTHER PRIMARY TEXT: Compare the definition of subgroup given here, with the definition in your primary text

## Definition

### Definition in terms of closure under binary operation

This definition of subgroup corresponds to the textbook definition of group.

Let be a group. A subset of is termed a **subgroup** if the following two conditions hold:

- Whenever belong to , the product belongs to .
- With this induced multiplication, becomes a group in its own right (i.e., it has an identity element, and every element has a two-sided inverse). Note that associativity in follows automatically from associativity in .

### The universal algebraic definition

This definition of subgroup corresponds to the universal algebraic definition of group.

Let be a group. A subset of is termed a **subgroup** if all the three conditions below are satisfied:

- Whenever belong to , so does (here denotes the product of the two elements)
- belongs to (where denotes the identity element)
- Whenever belongs to , so does (the multiplicative inverse of )

### Definition via the subgroup condition

The equivalence of this definition with the earlier one is often called the **subgroup condition**. `For full proof, refer: Sufficiency of subgroup condition`

It has two forms (left and right):

- A subset of a group is termed a subgroup if it is nonempty and is closed under the left quotient of elements. In other words, a subset of a group is termed a subgroup if and only if is nonempty and whenever
- A subset of a group is termed a subgroup if it is nonempty and is closed under the right quotient of elements. In other words, a subset of a group is termed a subgroup if and only if is nonempty and whenever

## Notation

If is a subgroup of , we typically write or . Some people also write , but the latter notation is typically used for arbitrary subsets that need not be subgroups.

If is not equal to the whole of , we say that is a proper subgroup of , and this is sometimes denoted by or .

## Examples

### Examples in abelian groups

If we consider the abelian group (reals under addition) then the group of integers is a subgroup of this group. Similarly, the group of rational numbers () is an example of a subgroup of the group of reals.

On the other hand, the set of positive integers is *not* a subgroup of the group of integers, although it is closed under the group operation. This is because the additive inverse (or negative) of a positive integer isn't a positive integer.

PONDER (WILL BE EXPLORED LATER IN THE TOUR):

- Over why the different definitions of subgroup are equivalent.
- Over what parts of the definition of group are needed to prove the equivalence of definitions
WHAT'S MORE: Some more definitions, examples, and general information about subgroups. Some of it may use terminology that you haven't encounutered so far; ignore those parts.

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)[SHOW MORE]PREVIOUS: Abelian group|UP: Introduction one (beginners)|NEXT: Trivial group

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part