Subgroup

Definition

The universal algebraic definition

Let $G$ be a group. A subset $H$ of $G$ is termed a subgroup if:

• Whenever $a,b$ belong to $H$, so does $ab$
• Whenever $a$ belongs to $H$, so does $a^{-1}$
• $e$ belongs to $H$

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 $H_{1}$ ≤ $G_{1}$ and a subgroup $H_{1}$ ≤ $G_{2}$, 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.

Metaproperties

Intersection-closedness

An arbitrary intersection of subgroups is a subgroup. For full proof, refer: Intersection of subgroups is subgroup Thus, given any subset of a group, it makes sense to talk of the smallest subgroup containing that subset.

Join-closedness

Given any subset, we can talk of the subgroup generated by that subset. One way of viewing this is as the intersection of all subgroups containing that subset. Another way of viewing it is as the set of all elements in the group that can be expressed using elements of the subset, and the group operations.

Hence, in particular, given a family of subgroups, we can talk of the subgroup generated by them, as simply the subgroup generated by their union. This is the smallest subgroup containing all of them.

Transitivity

Any subgroup of a subgroup is again a subgroup. This follows directly from any of the equivalent definitions of subgroup.

Trimness

There are two extreme kinds of subgroups: the trivial subgroup, which comprises only the identity element, and the whole group, which comprises all elements.

Intermediate subgroup condition

The property of being a subgroup satisfies the intermediate subgroup condition. That is, if $H\leq G$ is a subgroup and $K$ is a subgroup of $G$ containing $H$, then $H$ is a subgroup of $K$ (not merely a subset).

Image condition

The image of a subgroup under any homomorphism of groups is again a subgroup.

Inverse image condition

The inverse image of a subgroup under any homomorphism of groups is again a subgroup.

ACU-closedness

The union of any ascending chain of subgroups is again a subgroup. In fact, it is precisely the subgroup generated by the members of the ascending chain.