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Appendix: Review of Basic Algebra
Definition A.2 Let (G, ∗) be a group and H ⊆ G. We say that H is a subgroup of
G if:
(SG1) e ∈ H , where e is the identity of G.
(SG2) If a, b ∈ H , then a ∗ b ∈ H .
(SG3) If a ∈ H , then a
−1
∈ H .
A characterization of subgroup is presented in the sequence.
Proposition A.2 Let (G, ∗) be a group. A subset H ⊆ G is a subgroup of G if and
only if H is nonempty and for all x, y ∈ H , it follows that x ∗ y
−1
∈ H.
Proof See Proposition 2.30 in [137].
An interesting and much utilized class of subgroups is the class of normal subgroups.
Definition A.3 Let (G, ∗) be a group and H be a subgroup of G. We say that H is
a normal subgroup of G if, for all h ∈ H and g ∈ G, imply g ∗ h ∗ g
−1
∈ H .
An important feature of normal subgroups is that such groups allow us to define
quotient groups.
Definition A.4 Assume that (G, ∗) is a group and H is a subgroup of G. For all
g ∈ G, we define the left coset of g by H as the set g ∗ H = {g ∗ h|h ∈ H }.
Proposition A.3 Let (G, ∗) be a group, H ⊆ G subgroup of G and a, b ∈ G. Then
the following hold:
(i) a ∗ H = b ∗ H if and only if b
−1
∗ a ∈ H . In particular, a ∗ H = H if and only
if a ∈ H.
(ii) If (a ∗ H ) ∩ (b ∗ H ) = ∅, then a ∗ H = b ∗ H.
Proof The proofs follow because the relation on G, defined by a ∼ b ⇐⇒ b
−1
∗ a ∈
H , is an equivalence relation whose equivalence classes are the left cosets.
If X, Y are nonempty subsets of G, we define the set
X ∗ Y = {x ∗ y|x ∈ X and y ∈ Y }.
Theorem A.1 Let (G, ∗) be a group and H ⊆ G be a normal subgroup of G. Let
G/H be the family of all the left cosets of H . Then, for all a, b ∈ G, one has (a ∗
H )(b ∗ H ) = (a ∗ b)H , and G/H is a group under this operation.
Proof See Theorem 2.67 in [137].
The group G/H is said to be the quotient group G mod H .
In the sequence we define the concept of ring.