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

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 $a,b \in H$

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 definitions

Further information: Equivalence of definitions of subgroup, Sufficiency of subgroup criterion

The equivalence of the two definitions (the definition in terms of closure under binary operation and the universal algebraic definition) relies on the following two facts:

1. If a subset of a group is closed under multiplication and has an identity element, then that identity element must equal the identity element of the group (this relies on the cancellation property in groups).
2. If a subset of a group allows for multiplicative inverses, then the multiplicative inverses inside the subset are the same as the multiplicative inverses inside the whole group

The equivalence with the definition arising from the subgroup criterion is based on a short and elegant argument, refer sufficiency of subgroup criterion.

The equivalence with the definition in terms of injective homomorphism relies on viewing the subgroup as a group in its own right and its inclusion in the whole group as an injective homomorphism.

Equivalence of subgroups

Given a subgroup $H_1 \le G_1$ and a subgroup $H_2 \le G_2$, 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 $H_2$ (subgroups equivalent in this sense are termed automorphic subgroups or automorphs -- sometimes, stronger notions of equivalence, such as being conjugate subgroups, are also useful).

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

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.

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.

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 $H \le G$ 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.

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.

References

Textbook references

• Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 22, Exercise 26 (definition introduced in exercise), and Page 46 (formal definition)
• Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 2 (definition introduced in paragraph)
• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 8 (definition introduced in paragraph)
• An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 46 (definition introduced in paragraph)
• Algebra by Serge Lang, ISBN 038795385X, ^{More info}, Page 9 (definition introduced in paragraph)
• A First Course in Abstract Algebra (6th Edition) by John B. Fraleigh, ISBN 0201763907, ^{More info}, Page 66 (formal definition)
• Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 31, Definition 2.4 (formal definition)
• Contemporary Abstract Algeba by Joseph Gallian, ISBN 0618514716, ^{More info}, Page 57
• Topics in Algebra by I. N. Herstein, ^{More info}, Page 37 (formal definition)
• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 44, Section 2 (formal definition)

External links

Definition links

• Wikipedia page
• Planetmath page
• Mathworld page
• Springer Online Reference page