Question

prove that cancellation laws holds in a group

Answer 1

Proof: Let a−1∈G denote the inverse of a under ∗. Suppose that a∗b=a∗c. Then:
(1)
a∗b=a∗c(a−1∗a)∗b=(a−1∗a)∗ce∗b=e∗cb=c
Similarly, suppose now that b∗a=c∗a. Then:
(2)
b∗a=c∗a(b∗a)∗a−1=(c∗a)∗a−1b∗(a∗a−1)=c∗(a∗a−1)b∗e=c∗eb=c■

Answer 2

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Solution :

To prove :

If a , b , c are any three arbitrary elements of a group G , then

1. ab = ac → b = c (left cancellation law)
2. ba = ca → b = c (right cancellation law)

Proof :

Let a⁻¹ ∈ G be the inverse element of a ∈ G , then we have a⁻¹a = aa⁻¹ = e , where e is the identity element in G .

Now , ab = ac

→ a⁻¹(ab) = a⁻¹(ac)

→ (a⁻¹a)b = (a⁻¹a)c

→ eb = ec

→ b = c

Also , ba = ca

→ (ba)a⁻¹ = (ca)a⁻¹

→ b(aa⁻¹) = c(aa⁻¹)

→ be = ce

→ b = c

Hence proved .