Discrete Mathematical Structures by Pearson

Algebraic Structures 7-33

7.9.1 Congruence Relation in Groups

Let H be a subgroup of group G. Define congruence relation in G by a ≡ b (mod H), a, b ∈ G.

if b ∈ aH, that is, b = ah

1

for some h

1

∈ H

This congruence relation is an equivalence relation in G as can be seen from the following:

1. Reflexivity a ≡ a(mod H) since a = ae (e ∈ H)

2. Symmetry a ≡ b(mod H) ⇒ b ≡ a (mod H)

If a ≡ b(mod H), then b = ah

1

(h

1

∈ H, b ∈ aH)

∴ a = bh

1

-1

Therefore, a ∈ bH, which shows that b ≡ a(mod H).

3. Transitivity a ≡ b(mod H), b ≡ c(mod H) ⇒ a ≡ c(mod H)

Now

a ≡ b(mod H) ⇒ b ∈ aH, that is, b = ah

1

(h

1

∈ H)

and

b ≡ c(mod H) ⇒ c ∈ bH, that is, c = bh

2

(h

2

∈ H)

Hence,

c = bh

2

= (ah

1

) h

2

= a(h

1

h

2

) = ah

3

(h

1

h

2

= h

3

∈ H)

That is, a ≡ c(mod H).

This equivalence relation ...