1.The set Z = { . . .,n, . . .,−1, 0, 1, . . . , n, . . .}of integers is an abelian group under addition, with identity element 0. However, Z ^∗ = Z − {0} is not a group under multiplication.

2.The set Q of rational numbers (fractions p/q with p, q ∈ Z and q ≠ 0) is an abelian group under addition, with identity element 0. The set Q∗ = Q− {0} is also an abelian group under multiplication, with identity element 1.

3.Given any nonempty set S, the set of bijections f : S → S, also called permutations  of S, is a group under function composition (i.e., the multiplication of f and g is the composition g ◦ f), with identity element the identity function id_S. This group is not abelian as soon as S has more than two elements. The permutation group of the set S = {1, . . . , n} is often denoted Gn and called the symmetric group on n elements.

4.For any positive integer p ∈ N, define a relation on Z, denoted m ≡ n (mod p), as follows:

m ≡ n (mod p) iff m − n = kp for some k ∈ Z.

The reader will easily check that this is an equivalence relation, and, moreover, it is compatible with respect to addition and multiplication, which means that if m₁ ≡ n₁ (mod p) and m2 ≡ n₂ (mod p), then m₁ + m₂ ≡ n₁ + n₂ (mod p) and m₁m₂ ≡ n₁n₂ (mod p). Consequently, we can define an addition operation and a multiplication operation of the set of equivalence classes (mod p):

[m] + [n] = [m + n]

and

[m] · [n] = [mn].

The reader will easily check that addition of residue classes (mod p) induces an abelian group structure with [0] as zero. This group is denoted Z/pZ.

5.The set of n×n invertible matrices with real (or complex) coefficients is a group under matrix multiplication, with identity element the identity matrix I_n. This group is called the general linear group and is usually denoted by GL(n, R) (or GL(n, C)).

6.The set of n × n invertible matrices A with real (or complex) coefficients such that det(A) = 1 is a group under matrix multiplication, with identity element the identity matrix I_n. This group is called the special linear group and is usually denoted by SL(n, R) (or SL(n, C)).

7.The set of n × n matrices Q with real coefficients such that

QQ^T = Q T Q = I_n is a group under matrix multiplication, with identity element the identity matrix I_n; we have Q⁻¹ = Q^T . This group is called the orthogonal group and is usually denotedby O(n).

8.The set of n × n invertible matrices Q with real coefficients such that

QQ^T = Q ^T Q = I_n and det(Q) = 1

is a group under matrix multiplication, with identity element the identity matrix I_n; as in (6), we have Q⁻¹ = Q^T . This group is called the special orthogonal group or rotation group and is usually denoted by SO(n).

The groups in (5)–(8) are nonabelian for n ≥ 2, except for SO(2) which is abelian (but O(2)is not abelian).

It is customary to denote the operation of an abelian group G by +, in which case the inverse a ⁻¹ of an element a ∈ G is denoted by −a.

The identity element of a group is unique. In fact, we can prove a more general fact:

2024-10-08

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