Appendix
Review of Basic Algebra
Here, we present some basic definitions of algebra, in order to make this book selfconsistent. The first definition is that of group.
Definition A.1 Let G be a set and ∗ : G × G −→ G a binary operation on G. We
say that (G, ∗) is a group if the following conditions hold:
(G1) There exists an element e ∈ G, called identity, such that, for all x ∈ G, e ∗
x = x ∗ e = x.
(G2) The operation ∗ is associative: for all x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z.
(G3) For all x ∈ G, there exists x
∈ G such that x ∗ x
= x
∗ x = e. The element
x
is an inverse of x.
If, in addition, the operation satisfies the commutativity, that is,
(G4) For all x, y ∈ G, x ∗ y = y ∗ x,
then the group is said to be abelian (this name was coined due to Niels Henrik Abel).
Groups satisfy several properties.
Proposition A.1 Let (G, ∗) be a group. Then the following hold:
(I)
The identity is unique, i.e., there exists a unique element e ∈ G such that, for
all x ∈ G, x ∗ e = e ∗ x = x.
(II)
For every x ∈ G, its inverse is unique. More precisely, for every x ∈ G, there
exists a unique x
−1
∈ G such that x ∗ x
−1
= x
−1
∗ x = e.
(III) The cancelation laws hold: assume that x, a, b ∈ G; if either x ∗ a = x ∗ b
or a ∗ x = b ∗ x, then it follows that a = b.
(IV) For all x ∈ G, (x
−1
)
−1 = x.
(V) For all x, y ∈ G, (x ∗ y)
−1
= y
−1
∗ x
−1 .
(VI) Let a, b ∈ G. Then the equation a ∗ x = b has a unique solution in G, namely,
x 0 = a
−1
∗ b.
Proof The proofs are routine and can be found in textbooks of algebra; see, for
instance, [137].
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1
213
Review of Basic Algebra
Here, we present some basic definitions of algebra, in order to make this book selfconsistent. The first definition is that of group.
Definition A.1 Let G be a set and ∗ : G × G −→ G a binary operation on G. We
say that (G, ∗) is a group if the following conditions hold:
(G1) There exists an element e ∈ G, called identity, such that, for all x ∈ G, e ∗
x = x ∗ e = x.
(G2) The operation ∗ is associative: for all x, y, z ∈ G, x ∗ (y ∗ z) = (x ∗ y) ∗ z.
(G3) For all x ∈ G, there exists x
∈ G such that x ∗ x
= x
∗ x = e. The element
x
is an inverse of x.
If, in addition, the operation satisfies the commutativity, that is,
(G4) For all x, y ∈ G, x ∗ y = y ∗ x,
then the group is said to be abelian (this name was coined due to Niels Henrik Abel).
Groups satisfy several properties.
Proposition A.1 Let (G, ∗) be a group. Then the following hold:
(I)
The identity is unique, i.e., there exists a unique element e ∈ G such that, for
all x ∈ G, x ∗ e = e ∗ x = x.
(II)
For every x ∈ G, its inverse is unique. More precisely, for every x ∈ G, there
exists a unique x
−1
∈ G such that x ∗ x
−1
= x
−1
∗ x = e.
(III) The cancelation laws hold: assume that x, a, b ∈ G; if either x ∗ a = x ∗ b
or a ∗ x = b ∗ x, then it follows that a = b.
(IV) For all x ∈ G, (x
−1
)
−1 = x.
(V) For all x, y ∈ G, (x ∗ y)
−1
= y
−1
∗ x
−1 .
(VI) Let a, b ∈ G. Then the equation a ∗ x = b has a unique solution in G, namely,
x 0 = a
−1
∗ b.
Proof The proofs are routine and can be found in textbooks of algebra; see, for
instance, [137].
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1
213