# Tour:Subgroup

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General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
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 operation and the universal algebraic definition of 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 subgroup means

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 $G$ be a group. A subset $H$ of $G$ is termed a subgroup if the following two conditions hold:

• Whenever $a,b$ belong to $H$, the product $ab$ belongs to $H$.
• With this induced multiplication, $H$ 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 $H$ follows automatically from associativity in $G$.

### The universal algebraic definition

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

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

• Whenever $a, b$ belong to $H$, so does $ab$ (here $ab$ denotes the product of the two elements)
• $e$ belongs to $H$ (where $e$ denotes the identity element)
• Whenever $a$ belongs to $H$, so does $a^{-1}$ (the multiplicative inverse of $a$)

### 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 $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $a^{-1}b \in H$ whenever $a,b \in H$
• 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 $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $ab^{-1} \in H$ whenever $a,b \in H$

## Notation

If $H$ is a subgroup of $G$, we typically write $H \le G$ or $G \ge H$. Some people also write $H \subseteq G$, but the latter notation is typically used for arbitrary subsets that need not be subgroups.

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

## Examples

### Examples in abelian groups

If we consider the abelian group $(\R,+)$ (reals under addition) then the group of integers $(\mathbb{Z},+)$ is a subgroup of this group. Similarly, the group of rational numbers ($\mathbb{Q},+)$) 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.