Subgroup

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article defines a property of subsets of groups
View other properties of subsets of groups|View properties of subsets of abelian groups|View subgroup properties

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 $a b$ (here $a b$ 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 $a b^{- 1} \in H$ 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 $σ$ 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 $R$ (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 ${1, 2, 3, \dots, n}$ . 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.

Properties

Subgroup properties

Further information: subgroup property Given a group and a subgroup thereof, we want answers to various questions about how the subgroup sits inside the group. These answers are encoded in various ways. One of these is by checking whether the subgroup satisfies a particular subgroup property. A subgroup property is something that takes as input a group and subgroup and outputs true/false; moreover, the answer should be the same for equivalent group-subgroup pairs.

Category:Subgroup properties is a complete list of subgroup properties; Category:Pivotal subgroup properties is a list of important ones.

Note that the property of being a subgroup is itself a subgroup property; in logical terms, it is the tautology subgroup property: the one that's always true.

Metaproperties

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving 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

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving 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

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about 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

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

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

Inverse image condition

This subgroup property satisfies the inverse image condition. In other words, the inverse image under any homomorphism of a subgroup satisfying the property also satisfies the property. In particular, this property satisfies the transfer condition and intermediate subgroup condition.

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

Template:ACU-closed

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.