Definition (Page 29). A group is a set G together with an operation ∗ on G such that each of the following axioms is satisfied:

  1. Associative. For all a,b,c∈G, a∗(b∗c)=(a∗b)∗c.
  2. Identity. There is an e∈G such that for each a∈G a∗e=e∗a=a.
  3. Inverse. For each a∈G there is an element b∈G such that a∗b=b∗a=e.

Guevara Note (Page 29). (Closure) By definition, to be a group, G must be closed with respect to the operation ∗. Therefore, if G is a group with operation ∗ and H⊂G but there are a,b∈H such that a∗b∉H, then H cannot be a group since it is not closed with respect to ∗. In such a case, we still have a∗b∈G since G is a group.

Note (Page 29). Sometimes we refer to a group only by it's set name when the operation is implied. For example, the group of integers refers to the integers with addition.

Guevara Note (Page 29). The identity condition implies a group G is nonempty, since it must contain e, by definition.

Example 5.1. The even integers with addition is a group.

Example 5.2. The positive integers with addition is not a group.

Example 5.3. The set {0} with addition is a group.

Example 5.4. The positive rational numbers with multiplication is a group.

Example 5.5. See tables 5.1 and 5.2 in the text.

Example 5.6. If S is a nonempty set, then the set of all invertible mappings in M(S) with composition is a group. This is a restatement of theorem Theorem 4.1(b).

Guevara Example (Page 30). The set M(S) is not a group, since there is an α∈M(S) that has no inverse.

Example 5.7. The set of all rotations of the plane about a fixed point in the plane with composition is a group. This is a restatement of Example 4.1.

Example 5.8. From Example 4.2, the set of all mappings αa,b:R→R where a,b∈R, a≠0 and αa,b(x)=ax+b for each x∈R, with composition is a group.

Example 5.9. Examples of groups of matrices Problems 5.16 and 5.17 and Appendix D. Such groups are very important.

Theorem 5.1. If G with ∗ is a group then

  1. Unique Identity. The identity element of G is unique. That is, if e∗a=a∗e=a for each a∈G and f∗a=a∗f=a for each a∈G then e=f. Thus, we may speak of the identify of the group.
  2. Unique Inverse. Each element in a group has a unique inverse. That is, if a,x,y∈G and e is the identity of G, and a∗x=x∗a=e and a∗y=y∗a=e then x=y. Thus, we may speak of the inverse of a group element and write a−1

Definition (Page 32). The inverse of a is written a−1. Thus, a∗a−1=a−1∗a=e. This notation aligns with exponent notation for groups discussed in Section 14 and is justified by the uniqueness of inverses. Other notation may be used if available. For example, in the group of integers we have a−1=−a.

Definition (Page 32). If two groups G and H are considered, the notation eG and eH may be used for their respective identities.

Definition (Page 32). The order of a group G denoted by |G| is the number of elements in G.