# Group Theory/Groups, subgroups and constructions

**Proof:** First note that when , then we may multiply by on the left using in order to get by using the identity axiom and to get left cancellation, and right cancellation is proved similarly. Then, we note that by multiplying the inverse axiom by on the left and using the identity axiom, we get and hence, since by the identity axiom, we may apply cancellation to infer , proving 2. Finally, .

**Definition (group)**:

A **group** is a set together with three operations, namely

- a nullary operation called the identity,
- a unary operation called inversion
- and a binary operation called the group operation (also often referred to simply as the "operation" of the group)

such that the following axioms are satisfied for all elements of :

- (identity axiom)
- (inverse axiom)
- (associativity axiom)

**Definition (abelian group)**:

An **abelian group** is a group , whose operation (which is usually denoted by ) is commutative.

**Proposition (elementary rules for computing in groups)**:

Let be a group with group operation and let . Then we have the following rules of computation:

- and ("cancellation laws")

Henceforth, we shall sometimes refer to the group operation of a group simply as the "operation".

**Definition (opposite group)**:

Let be a group, where denotes the operation of . Then the **opposite group** of is defined as the group that has the same underlying set as , but whose group operation is given by for (inversion and identity are carried over from ).

**Proposition (opposite groups are groups)**:

Whenever is a group, is also a group.

**Proof:** We have to check that the axioms for the given operations are satisfied. First note that and that by the rules that were derived above. Finally, .

**Definition (subgroup)**:

Let be a group. A subset is called a **subgroup** of iff together with the group law of it is itself a group (in particular, for all we must have .

Note: Often, the explicit notation for the group operation is omitted and the product of two elements is denoted solely by juxtaposition.

Subgroups with the inclusion map represent subobjects of a group.

**Proposition (subgroup criterion)**:

Let be a group and a subset. Then is a subgroup of if and only if .

**Proof:** Suppose first that is a subgroup. Then the condition follows immediately from being closed under inversion and the group law. On the other hand, if the condition is satisfied, setting shows that is closed under inversion, and setting shows that it is closed under the group law. Finally, setting shows that contains the identity. Associativity and the law that the identity must satisfy are automatically inherited from the group operation of .

## Exercises

edit- Make explicit the proof of right-cancellation ("right-cancellation" means ).
- Let be a group, and let be subgroups such that neither nor . Prove that is not a subgroup of .
- Let together with the operation , .
- Prove in detail that , together with the operation , is a group.
- Prove that in , there exists a subgroup which is not equal to with subgroups .