ABSTRACT ALGEBRAIC STRUCTURES OPERATIONS AND ...

Proof :
(1) ⇒ (2): If N is normal you have per definition that the associated left equivalence
relation is compatible with the group operation and as a result of this
calculation with the classes can be carried out by calculation with representatives,
which is what is stated in (2).
(2) ⇒ (3) Let x be an arbitrary member of G. Then by (2) we have that
(xN)(x −1 N) = (xx −1 )N = N. For an arbitrary member h ∈ N we then have
that xhx −1 e ∈ N. Therefore also xhx −1 ∈ N, and since h was arbitray it
follows that xNx −1 ⊆ N. The opposite inclusion is obvious.
(3) ⇒ (4): Exercise.
(4) ⇒ (1): Assuming (4) we must show that the left equivalence relation is
compatible with the group operation. Let a ∼ b and c ∼ d. Per definition this
means that b ∈ aN and d ∈ cN. So we may choose h, k ∈ N with b = ah and
d = ck. Since hc ∈ Nc and since Nc = cN per assumption it is possible to find
p ∈ N with hc = cp. Therefore bd = ahck = acpk and pk ∈ N. This shows that
bd ∈ acN which means that bd ∼ ac, establishing the compatibility. To show
that also inversion is compatible let a ∼ b, that is a ∈ bH = Hb. So you may
find h ∈ N with a = hb. Then a −1 = b −1 h −1 , which shows that a −1 ∈ b −1 H
and so a −1 ∼ b −1 which establishes the compatibility. These criteria are used
in X47 X48
108. Theorem: Commutativity guaranties normality
Any subgroup of a commutative group is normal.
Proof : Trivial.
109. Theorem: Same left and right coset in case of normality
If N is a normal subgroup , then the right equivalence relation is identical with
the left and so this relation is just said to be the equivalence relation induced
by N and is denoted by ∼N .
110. Theorem: Quotient group.
Let N be a normal subgroup of the group G. The induced quotient structure is
a group structure.
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