Subgroup

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.

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
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 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.

Subgroup properties
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
An arbitrary intersection of subgroups is a subgroup. Thus, given any subset of a group, it makes sense to talk of the smallest subgroup containing that subset.

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.

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

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

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).

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

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.

Textbook references

 * , Page 22, Exercise 26 (definition introduced in exercise), and Page 46 (formal definition)
 * , Page 2 (definition introduced in paragraph)
 * , Page 8 (definition introduced in paragraph)
 * , Page 46 (definition introduced in paragraph)
 * , Page 9 (definition introduced in paragraph)
 * , Page 66 (formal definition)
 * , Page 31, Definition 2.4 (formal definition)
 * , Page 57
 * , Page 37 (formal definition)
 * , Page 44, Section 2 (formal definition)