Theory of Statistics - George Mason University

0.0 Some Basic Mathematical Concepts 625
is a group. Notice that the G may contain elements that are in neither G1 nor
G2.
0.0.3.3 Homomorphisms
It is often of interest to consider relationships between two groups. The simplest
and most useful is a morphism, or homomorphism. (The two words are
synonymous; I will generally use the latter.) Given two groups G = (G, ◦) and
G ∗ = (G ∗ , ⋄), a homomorphism from G to G ∗ is a function f from G to G ∗
such that for g1, g2 ∈ G,
f(g1 ◦ g2) = f(g1) ⋄ f(g2).
Often in applications, G and G ∗ are sets of the same kind of objects, say
functions for example, and the operations ◦ and ⋄ are the same, say function
composition.
If the homomorphism from G to G ∗ is a bijection, then the homomorphism
is called an isomorphism, and since it has an inverse, we say the two groups
are isomorphic. Isomorphic groups of transformations are the basic objects
underlying the concept of equivariant and invariant statistical procedures.
0.0.3.4 Structures with Two Binary Operators
Often it is useful to define two different binary operators, say + and ◦ over
the same set. An important type of relationship between the two operators is
called distributivity:
distributivity We say ◦ is distributive over + in S iff
x, y, z ∈ S =⇒ x ◦ (y + z) = (x ◦ y) + (x ◦ z) ∈ S.
Another very useful algebraic structure is a field, which is a set and two
binary operations (S, +, ◦) with special properties.
Definition 0.0.3 (field)
Let S be a set with two distinct elements and let + and ◦ be binary operations.
The structure (S, +, ◦) is called a field if the following conditions hold.
(f1) x1, x2 ∈ S ⇒ x1 + x2 ∈ S (closure of +);
(f2) ∃ 0 ∈ S ∋ ∀x ∈ S, 0 + x = x (identity for +);
(f3) ∀ x ∈ S ∃ − x ∈ S ∋ −x ◦ x = e (inverse wrt +);
(f4) x1, x2 ∈ S ⇒ x1 + x2 = x2 + x1 (commutativity of +);
(f5) x1, x2, x3 ∈ S ⇒ x1 ◦ (x2 ◦ x3) = (x1 ◦ x2) ◦ x3 (associativity of +).
(f6) x1, x2 ∈ S ⇒ x1 ◦ x2 ∈ S (closure of ◦);
(f7) ∃ 1 ∈ S ∋ ∀x ∈ S, 1 ◦ x = x (identity for ◦);
(f8) ∀ x = 0 ∈ S ∃ x −1 ∈ S ∋ x −1 ◦ x = 1 (inverse wrt ◦);
(f9) x1, x2 ∈ S ⇒ x1 ◦ x2 = x2 ◦ x1 (commutativity of ◦);
Theory of Statistics c○2000–2013 James E. Gentle