Tour:Subgroup

This article adapts material from the main article: Subgroup

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Given below is the definition of subgroup. A subgroup of a group is, roughly speaking, a subset that's also a group. The definition of subgroup is given in a number of equivalent ways below. Before proceeding further, make sure that you fully understand why all the definitions are equivalent, and why they tally with whatever definitions you see in textbooks.
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Definition

The universal algebraic definition

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)
• Whenever $a$ belongs to $H$, so does $a^{-1}$ (the multiplicative inverse of $a$)
• $e$ belongs to $H$ (where $e$ denotes the identity element)

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 $a,b\in H$

Definition in terms of injective homomorphisms

A subgroup of a group can also be defined as another abstract group along with an injective homomorphism (or embedding) from that abstract group to the given group. Here, the other abstract group can be naturally identified via its image under the homomorphism, which is the subgroup in a more literal sense.

Often, when we want to emphasize the subgroup not just as an abstract group but in its role as a subgroup, we use the term embedding and think of it as an injective homomorphism.

Equivalence of subgroups

Given a subgroup $H_{1}\leq G_{1}$ and a subgroup $H_{2}\leq G_{2}$, we say that these two subgroups are equivalent if there is an isomorphism $\sigma$ from $G_{1}$ to $G_{2}$ such that $H_{1}$ maps to $H_{2}$ under that isomorphism.

In particular, if $G_{1}=G_{2}=G$, then $H_{1}$ and $H_{2}$ are equivalent as subgroups if there is an automorphism of $G$ under which $H_{1}$ maps to $H_{2}$.

This notion of equivalence of subgroups is important when dealing with and defining the notion of subgroup property.

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

The universal algebraic definition

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)
• Whenever $a$ belongs to $H$, so does $a^{-1}$ (the multiplicative inverse of $a$)
• $e$ belongs to $H$ (where $e$ denotes the identity element)

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 $a,b\in H$

Definition in terms of injective homomorphisms

A subgroup of a group can also be defined as another abstract group along with an injective homomorphism (or embedding) from that abstract group to the given group. Here, the other abstract group can be naturally identified via its image under the homomorphism, which is the subgroup in a more literal sense.

Often, when we want to emphasize the subgroup not just as an abstract group but in its role as a subgroup, we use the term embedding and think of it as an injective homomorphism.

Equivalence of subgroups

Given a subgroup $H_{1}\leq G_{1}$ and a subgroup $H_{2}\leq G_{2}$, we say that these two subgroups are equivalent if there is an isomorphism $\sigma$ from $G_{1}$ to $G_{2}$ such that $H_{1}$ maps to $H_{2}$ under that isomorphism.

In particular, if $G_{1}=G_{2}=G$, then $H_{1}$ and $H_{2}$ are equivalent as subgroups if there is an automorphism of $G$ under which $H_{1}$ maps to $H_{2}$.

This notion of equivalence of subgroups is important when dealing with and defining the notion of subgroup property.

Examples

Examples in Abelian groups

If we consider the Abelian group $\mathbb {R}$ (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.

Examples in non-Abelian groups

Consider the group of all permutations of the set of elements $\{1,2,3,\ldots ,n\}$. This is termed the symmetric group on $n$ elements. The group of permutations that fix $n$ is a subgroup of this group.

Subgroups usually arise as elements of the group satisfying some additional conditions, where that condition is preserved under taking inverses, preserved under multiplication, and satisfied by the identity element.